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Solutions to Quartiles, Deciles and Percentiles Problems

Percentiles, quartiles and deciles, what are percentiles, example 1: finding the percentile value, what are deciles, problem 1: finding deciles and percentiles, solution to problem 1, what are quartiles.

If you’ve ever taken a standardized test, you will have used the word “percentile” before. It’s association with what many students consider the bane of their existence - official exams - isn’t doing the concept any favours. However, percentiles might be more important than many people realize - involved in everything from the apps on your phone to the algorithms that choose which songs you’re most likely to enjoy.

In this section, you’ll learn everything you need to know about what percentiles are, how to calculate them, and why they’re important in statistics.

The simple definition for a percentile is that it indicates the number at which a certain percentage of data falls below. As you learned in previous sections, there are two types of measurements in descriptive statistics: measures of central tendency and variability.

Percentiles are one version of measuring the variability within a data set. A percentile is an important measure because it can help you understand a certain data set better than simple means, modes or medians can.

The easiest way to understand why is to look at an example. You have a group of test scores out of 100 points from a class, following the table below.

You scored 50 points. At first glance, 50 out of 100 points may seem like a disappointing grade - for many classes, it would also be considered at the point of failure. However, calculating the percentile, you are at the 90th percentile. In other words, 90% of students scored lower than you did.

To calculate the percentile, your data should be ordered from least to greatest, similar to taking the median. Next, take the number for which you’d like to calculate the percentile for, in our case 50, and count what position its in.

You can be in a situation where you want to find the value corresponding to a certain percentile. Taking our example above, you want to find the 70th percentile, or the score at which 70% of students scored below.

To do this, we take

0.70*10 = 9

The index gives you the observation number for which your 70th percentile is located. If it has a decimal, round to the nearest whole number. Here, the index 7 means that the 7th observation in our data set is the score at the 70th percentile. Counting from the lowest to the highest score, we reach the 7th observed value: a score of 40, which is the 70th percentile for our data.

If we wanted to find the median , we can also use percentiles. For odd numbers, the median is:

\dfrac{n}{2} = index

In this case, since we have an even amount of numbers, we take the average of two indices,

index \thickspace 1 = \dfrac{n}{2}

index \thickspace 2 = \dfrac{n}{2}  + 1

Here, we get

\dfrac{10}{2} = 5

\dfrac{10}{2} + 1 = 6

Meaning the, the median is the average of the scores found at the fifth and sixth values in our ordered data set,

\dfrac{34+36}{2} = 34 \thickspace points

This means that 50% of students scored below 34 points and the other 50% of students scored above 34 points.

Deciles are a form of percentiles that split the data up into groups of 10%. Meaning, every decile contains 10% of the data. To find the decile, first order the data from least to greatest. Then, divide the data by 10. This indicates the number of observed values within each decile.

Using our previous example, we divide our data into 10 groups, each containing 10% of the data. This can be visualized in the data above. Because our n is equal to 10, each decile contains only 1 score.

The 1st decile = 15 . This score, at the 1st decile, is at the 10th percentile. Meaning, 10% of students scored below this number. This doesn’t really have much meaning here because there’s only 1 value at the 1st decile - however, it can be interpreted for data sets with larger sample sizes.

The 6th decile = 36 . This score, at the 6th decile, is at the 60th percentile, meaning that 60% of students scored below this number.

Using the first example, fill in the rest of the table with the corresponding deciles and percentiles.

You should obtain the following result:

Similar to deciles, quartiles are a form of percentiles. While deciles split the data into 10 “buckets,” quartiles split them into quarters . A good way of remembering this is that “deci” means a tenth, whereas quartile sounds similar to quarter, which is a fourth.

Splitting our data set into quarters, gives us the following.

Which is easier to understand when visualized:

From the image above, we can see that each quartile, or “bucket” contains 25% of our data. The score 24 is at the 25th percentile, which means that 25% of students scored below this score. The reason why this is at the 25th percentile and not the 30th percentile this time is because half of the score belongs to the first quartile and half belongs to the second.

The second quartile is also known as the median, which, as we calculated earlier, is 34 points. Quartile 3 is the 75th percentile, which means that at 41 points, 25% students scored above and 75% of students scored below this number.

The properties of quartiles are noted below.

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Decile is a method that is used to divide a distribution into ten equal parts. When data is divided into deciles a decile rank is assigned to each data point in order to sort the data into ascending or descending order. A decile has 10 categorical buckets while a quartile has 4 and a percentile has 100.

The concept of a decile is used widely in the field of finance and economics to perform the analysis of data. It can be used to check the performance of a portfolio in the field of finance. In this article, we learn more about a decile, its definition, rank, and see associated examples on calculating the decile value.

What is Decile?

Decile, percentile , quartile , and quintile are different types of quantiles in statistics. A quantile refers to a value that divides the observations in a sample into equal subsections. There will always be 1 lesser quantile than the number of subsections created.

Decile Definition

Decile is a type of quantile that divides the dataset into 10 equal subsections with the help of 9 data points. Each section of the sorted data represents 1/10 of the original sample or population. Decile helps to order large amounts of data in the increasing or decreasing order. This ordering is done by using a scale from 1 to 10 where each successive value represents an increase by 10 percentage points.

Decile Class Rank

To split the given data and order it according to some specified metric, statisticians use the decile rank also known as decile class rank. Once the given data is divided into deciles then each subsequent data set is assigned a decile rank. Each rank is based on an increase by 10 percentage points and is used to order the deciles in the increasing order. The 5th decile of a distribution will give the value of the median.

Decile Formula

The decile formulas can be used to calculate the deciles for grouped and ungrouped data . When data is in its raw form it is known as ungrouped data. When this data is sorted and organized then it forms grouped data. These are given as follows:

Decile Formula for ungrouped data: D(x) = Value of the \(\frac{x(n+1)}{10}\)th term in the data set.

x is the value of the decile that needs to be calculated and ranges from 1 to 9. n is the total number of observations in that data set.

Decile Formula for grouped data: D(x) = \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

l is the lower boundary of the class containing the decile given by (x × cf) / 10, cf is the cumulative frequency of the entire data set, w is the size of the class, N is the total frequency, C is the cumulative frequency of the preceding class.

The next section will cover the steps for calculating a particular decile.

Decile Example

Suppose a data set consists of the following numbers: 24, 32, 27, 32, 23, 62, 45, 80, 59, 63, 36, 54, 57, 36, 72, 55, 51, 32, 56, 33, 42, 55, 30. The value of the first two deciles has to be calculated. The steps required are as follows:

• Step 1: Arrange the data in increasing order. This gives 23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, 80.
• Step 2: Identify the total number of points. Here, n = 23
• Step 3: Apply the decile formula to calculate the position of the required data point. D(1) = \(\frac{(n+1)}{10}\) = 2.4. This implies the value of the 2.4 th data point has to be determined. This will lie between the scores in the 2 nd and 3 rd positions. In other words, the 2.4 th data is 0.4 of the way between the scores 24 and 27
• Step 4: The value of the decile can be determined as [lower score + (distance)(higher score - lower score)]. This is given as 24 + 0.4 * (27 – 24) = 25.2
• Step 5: Apply steps 3 and 4 to determine the rest of the deciles. D(2) = \(\frac{2(n+1)}{10}\) = 4.8 th data between digit number 4 and 5. Thus, 30 + 0.8 * (32 – 30) = 31.6

Related Articles:

• Probability and Statistics
• Data Handling
• Summary Statistics

Important Notes on Decile

• A decile is a quantile that is used to divide a data set into 10 equal subsections.
• The 5 th decile will be the median for the dataset.
• The decile formula for ungrouped data is given as \(\frac{x(n+1)}{10}\)th term in the data set.
• The decile formula for grouped data is given by \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

Examples on Decile

• Example 1: Find the 6th and the 9th decile for the data in the above-mentioned example. Solution: The arranged data is 23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, 80 n = 23 D(6) = \(\frac{6(n+1)}{10}\) = 14.4 th data. This lies between 54 and 55. D(6) = 54 + 0.4 * (55 – 54) = 54.4 D(9) = \(\frac{9(n+1)}{10}\) = 21.6 th data. This lies between 63 and 72 D(9) = 63 + 0.6 * (72 – 63) = 68.4 Answer: D(5) = 54.4 and D(9) = 68.4
• Example 2: Find the median of the following data set using the concept of deciles. 55, 58, 61, 67, 68, 70, 74, 81, 82, 93, 20, 28, 29, 30, 36, 37, 39, 42, 53, 54 Solution: Arranging the data in increasing order 20, 28, 29, 30, 36, 37, 39, 42, 53, 54, 55, 58, 61, 67, 68, 70, 74, 81, 82, 93 The fifth decile is the median of the data set, thus, n = 20 D(5) = \(\frac{5(n+1)}{10}\) = 10.5 th data. This lies between D(5) = 54 + 0.5* (55 - 54) = 54.5

D(7) = \(\frac{7\times 100}{10}\) = 70th data in the cf column

This data lies in the 60 - 70 class

D(7) = \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\)

= \(60+\frac{10}{18}\left ( \frac{7 \times 100}{10} -52\right )\) = 70

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FAQs on Decile

What is a decile in statistics.

A decile in statistics is a method to divide the distribution into 10 equal parts by using 9 data points and assigning decile ranks to each point.

What is a Decile Class Rank?

Once the data set is sorted into deciles then a decile class rank is assigned to each point so as to arrange these deciles into increasing order.

What is the Decile Formula?

The decile formula for ungrouped data is determined by the value of the \(\frac{x(n+1)}{10}\) term. The formula for grouped data is \(l+\frac{w}{f}\left ( \frac{Nx}{10} -C\right )\).

How to Find the Value of the Median Using the Decile Formula?

The value of the 5 th decile represents the median . For ungrouped data the median will be given by D(5) = \(\frac{5(n+1)}{10}\) th term.

How to Interpret the First Decile?

The first decile is a point such that 90% of the data lies above it and 10% of the data lies below it. Similarly, the 2 nd decile is a point with 20% of data lying below it and 80% lying above it.

How to Calculate the Decile for Ungrouped Data?

The steps to calculate the decile for ungrouped data are as follows:

• Arrange the data in increasing order.
• Find the position of the decile using the formula D(x) = \(\frac{x(n+1)}{10}\) to check between which scores the decile will be.
• Find the value of the decile [lower score + (distance)(higher score - lower score)]

What are the Applications of Decile?

The concept of decile is widely used in the finance and economics industries to assess the performance of a mutual fund or a portfolio. It acts as a comparative number to measure the performance of an asset.

Definition, Solved Example Problems - Deciles | 11th Statistics : Chapter 5 : Measures of Central Tendency

Chapter: 11th statistics : chapter 5 : measures of central tendency.

Deciles are similar to quartiles. Quartiles divides ungrouped data into four quarters and Deciles divide data into 10 equal parts .

Example 5.33

Find the D 6  for the following data  

11, 25, 20, 15, 24, 28, 19, 21

Arrange in an ascending order

11,15,19,20,21,24,25,28

Example 5.34

Calculate  D 5  for the frequency distribution of monthly income of workers in a factory

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• Quartiles & Quantiles | Calculation, Definition & Interpretation

Quartiles & Quantiles | Calculation, Definition & Interpretation

Published on May 20, 2022 by Shaun Turney . Revised on June 21, 2023.

Quartiles are three values that split sorted data into four parts, each with an equal number of observations. Quartiles are a type of quantile .

• First quartile : Also known as Q1, or the lower quartile. This is the number halfway between the lowest number and the middle number.
• Second quartile : Also known as Q2, or the median . This is the middle number halfway between the lowest number and the highest number.
• Third quartile : Also known as Q3, or the upper quartile. This is the number halfway between the middle number and the highest number.

Table of contents

What are quartiles, how to find quartiles, lower and upper quartile calculator, step-by-step example, visualizing quartiles with boxplots, interpreting quartiles, what are quantiles, practice questions, other interesting articles, frequently asked questions about quartiles and quantiles.

Quartiles are a set of descriptive statistics . They summarize the central tendency and variability of a dataset or distribution.

Quartiles are a type of percentile . A percentile is a value with a certain percentage of the data falling below it. In general terms, k % of the data falls below the k th percentile.

• The first quartile (Q1, or the lowest quartile) is the 25th percentile, meaning that 25% of the data falls below the first quartile.
• The second quartile (Q2, or the median) is the 50th percentile, meaning that 50% of the data falls below the second quartile.
• The third quartile (Q3, or the upper quartile) is the 75th percentile, meaning that 75% of the data falls below the third quartile.

By splitting the data at the 25th, 50th, and 75th percentiles, the quartiles divide the data into four equal parts.

• In a sample or dataset, the quartiles divide the data into four groups with equal numbers of observations.
• In a probability distribution , the quartiles divide the distribution’s range into four intervals with equal probability.

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To find the quartiles of a dataset or sample, follow the step-by-step guide below.

• Count the number of observations in the dataset ( n ).
• Sort the observations from smallest to largest.
• Calculate n * (1 / 4).
• If n * (1 / 4) is an integer, then the first quartile is the mean of the numbers at positions n * (1 / 4) and n * (1 / 4) + 1.
• If n * (1 / 4) is not an integer, then round it up. The number at this position is the first quartile.
• Calculate  n * (2 / 4).
• If n * (2 / 4) is an integer, the second quartile is the mean of the numbers at positions n * (2 / 4) and n * (2 / 4) + 1.
•  If n * (2 / 4) is not an integer, then round it up. The number at this position is the second quartile.
• Calculate n * (3 / 4).
• If n * (3 / 4) is an integer, then the third quartile is the mean of the numbers at positions n * (3 / 4) and n * (3 / 4) + 1.
• If n * (3 / 4) is not an integer, then round it up. The number at this position is the third quartile.

There are multiple methods to calculate the first and third quartiles, and they don’t always give the same answers. There’s no universal agreement on the best way to calculate quartiles.

Imagine you conducted a small study on language development in children 1–6 years old. You’re writing a paper about the study and you want to report the quartiles of the children’s ages.

1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6, 6

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Boxplots are helpful visual summaries of a dataset. They’re composed of boxes, which show the quartiles, and whiskers, which show the lowest and highest observations.

To make a boxplot, you first need to calculate the five-number summary :

• This is the lowest observation. If you order the numbers in your dataset from lowest to highest, the minimum is the first number. The minimum is sometimes called the zeroth quantile.
• The first quartile
• The second quartile
• The third quartile
• This is the highest observation. If you order the numbers in your dataset from lowest to highest, the maximum is the last number. The maximum is sometimes called the fourth quantile.

With these five numbers, you can draw a boxplot:

This isn’t the only method of drawing boxplots. Although the box always indicates the quartiles, often the whiskers indicate 1.5 IQR from the Q1 and Q3.

1 , 1, 2, 2 , 2, 3, 3, 3 , 3, 4, 5 , 5, 6, 6

To create a boxplot, the first step is to calculate the five-number summary:

• The minimum: 1 year old
• The first quartile: 2 years old
• The second quartile: 3 years old
• The third quartile: 5 years old
• The maximum: 6 years old

These numbers determine the position of the box and whiskers on a number line:

Quartiles can give you useful information about an observation or a dataset.

Comparing observations

Quartiles are helpful for understanding an observation in the context of the rest of a sample or population. By comparing the observation to the quartiles, you can determine whether the observation is in the bottom 25%, middle 50%, or top 25%.

The second quartile, better known as the median , is a measure of central tendency . This middle number is a good measure of the average or most central value of the data, especially for skewed distributions or distributions with outliers .

Interquartile range

The distance between the first and third quartiles—the interquartile range (IQR)—is a measure of variability . It indicates the spread of the middle 50% of the data.

The IQR is an especially good measure of variability for skewed distributions or distributions with outliers . IQR only includes the middle 50% of the data, so, unlike the range , the IQR isn’t affected by extreme values.

The distance between quartiles can give you a hint about whether a distribution is skewed or symmetrical. It’s easiest to use a boxplot to look at the distances between quartiles:

Note that a histogram or skewness measure will give you a more reliable indication of skewness.

Identifying outliers

The interquartile range (IQR) can be used to identify outliers . Outliers are observations that are extremely high or low. One definition of an outlier is any observation that is more than 1.5 IQR away from the first or third quartile.

A quartile is a type of quantile.

Quantiles are values that split sorted data or a probability distribution into equal parts. In general terms, a q -quantile divides sorted data into q parts. The most commonly used quantiles have special names:

• Quartiles (4-quantiles) : Three quartiles split the data into four parts.
• Deciles (10-quantiles) : Nine deciles split the data into 10 parts.
• Percentiles (100-quantiles): 99 percentiles split the data into 100 parts.

There is always one fewer quantile than there are parts created by the quantiles.

How to find quantiles

To find a q -quantile, you can follow a similar method to that used for quartiles , except in steps 3–5, multiply n by multiples of 1/ q instead of 1/4.

For example, to find the third 5-quantile:

• Calculate n * (3 / 5).
• If  n * (3 / 5) is an integer, then the third 5-quantile is the mean of the numbers at positions  n * (3 / 5) and n * (3 / 5) + 1.
• If n * (3 / 5) is not an integer, then round it up. The number at this position is the third 5-quantile.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s t table
• Student’s t distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

You can use the QUARTILE() function to find quartiles in Excel. If your data is in column A, then click any blank cell and type “=QUARTILE(A:A,1)” for the first quartile, “=QUARTILE(A:A,2)” for the second quartile, and “=QUARTILE(A:A,3)” for the third quartile.

You can use the quantile() function to find quartiles in R. If your data is called “data”, then “quantile(data, prob=c(.25,.5,.75), type=1)” will return the three quartiles.

To find the quartiles of a probability distribution, you can use the distribution’s quantile function.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.

Variability is also referred to as spread, scatter or dispersion.

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Quartiles are the values that divide a list of numbers into quarters:

• Put the list of numbers in order
• Then cut the list into four equal parts
• The Quartiles are at the "cuts"

Example: 5, 7, 4, 4, 6, 2, 8

Put them in order: 2, 4, 4, 5, 6, 7, 8

Cut the list into quarters:

And the result is:

• Quartile 1 (Q1) = 4
• Quartile 2 (Q2), which is also the Median , = 5
• Quartile 3 (Q3) = 7

Sometimes a "cut" is between two numbers ... the Quartile is the average of the two numbers.

Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8

The numbers are already in order

In this case Quartile 2 is half way between 5 and 6:

Q2 = (5+6)/2 = 5.5

• Quartile 1 (Q1) = 3
• Quartile 2 (Q2) = 5.5

Interquartile Range

The "Interquartile Range" is from Q1 to Q3:

To calculate it just subtract Quartile 1 from Quartile 3 , like this:

The Interquartile Range is:

Q3 − Q1 = 7 − 4 = 3

Box and Whisker Plot

We can show all the important values in a "Box and Whisker Plot", like this:

A final example covering everything:

Example: Box and Whisker Plot and Interquartile Range for

4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11

Put them in order:

3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18

Cut it into quarters:

3, 4, 4 | 4, 7, 10 | 11, 12, 14 | 16, 17, 18

In this case all the quartiles are between numbers:

• Quartile 1 (Q1) = (4+4)/2 = 4
• Quartile 2 (Q2) = (10+11)/2 = 10.5
• Quartile 3 (Q3) = (14+16)/2 = 15
• The Lowest Value is 3 ,
• The Highest Value is 18

So now we have enough data for the Box and Whisker Plot :

And the Interquartile Range is:

Q3 − Q1 = 15 − 4 = 11

Statistics Made Easy

How to Calculate Quartiles for Grouped Data

Quartiles are values that split up a dataset into four equal parts.

You can use the following formula to calculate quartiles for grouped data:

• Q i = L + (C/F) * (iN/4 – M)

L : The lower bound of the interval that contains the i th quartile

C : The class width

F : The frequency of the interval that contains the i th quartile

N : The total frequency

M : The cumulative frequency leading up to the interval that contains the i th quartile

The following example shows how to use this formula in practice.

Example: Calculate Quartiles for Grouped Data

Suppose we have the following frequency distribution:

Now suppose we’d like to calculate the value at the third quartile (Q 3 ) of this distribution.

The value at the third quartile will be located at position (iN/4) in the distribution.

Thus, (iN/4) = (3*92/4) = 69.

The interval that contains the third quartile will be the 21-25 interval since 69 is between the cumulative frequencies of 58 and 70.

Knowing this, we can find each of the values necessary to plug into our formula:

• The lower bound of the interval is 21 .
• The class width is calculated as 25 – 21 = 4 .
• The frequency of the 21-25 class is 12
• The total cumulative frequency in the table is 92 .
• The cumulative frequency leading up to the 21-25 class is 58 .

We can then plug in all of these values into the formula from earlier to find the value at the third quartile:

• Q 3 = 21 + (4/12) * ((3)(92)/4 – 58)
• Q 3 = 24.67

The value at the third quartile is 24.67 .

You can use a similar approach to calculate the values for the first and second quartiles.

Additional Resources

The following tutorials provide additional information for working with grouped data:

How to Find Mean & Standard Deviation of Grouped Data How to Find the Mode of Grouped Data How to Find the Median of Grouped Data Grouped vs. Ungrouped Frequency Distributions

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Here you will learn about a quartile, including what a quartile is, how to find the lower quartile and upper quartile for a set of data, and why these measures are useful.

Students will first learn about quartiles as part of statistics and probability in 6 th grade.

What is a quartile?

A quartile divides an ordered data set into four equal parts (quarters). You can use abbreviations to label the quartiles: Q1, M or Q2, and Q3.

The first quartile, Q1, is \cfrac{1}{4} \: ( or 25 \%) of the way through the data – the lower quartile .

The second quartile, M or Q2, is \cfrac{1}{2} \: ( or 50 \%) of the way through the data – the median .

The third quartile, Q3 is \cfrac{3}{4} \: ( or 75 \%) of the way through the data – the upper quartile .

You can only find quartiles for quantitative (numerical) data sets, but they can be found for both discrete and continuous quantitative data sets.

Remember, discrete data is data that can only take certain values (counted data), whereas continuous data can take any value within a given range (measured data).

For a small data set, you can find quartiles by simply counting and finding the correct data points.

For example,

• Interquartile range

Finding the lower quartile and upper quartile for a data set allows you to examine the middle half of the data centralized around the median – this is useful if a data set contains a lot of outliers or extreme values.

The lower quartile and upper quartile can also be used to calculate the interquartile range (IQR), which is a measure of the variability or spread of the data.

I Q R=U Q-L Q or I Q R=Q 3-Q 1

The median and lower and upper quartiles, along with the minimum value and the maximum value of the data set, form a five-number summary of descriptive statistics for the data set.

This information can then be presented in a box plot (also known as a box and whisker plot or diagram), making it easy to compare with other sets of data.

Percentiles

Quartiles are one way of splitting data to analyze; you may see the word percentiles used when discussing data sets in different contexts, such as news reports or test scores.

For example, you might see something about ‘the top 10 \% of test scores’ – to calculate this, you would find the 90 th percentile using a similar method to that for quartiles.

There is a link between quartiles and percentiles, as follows:

• First/lower quartile (Q1) - 25 th percentile
• Second quartile/median (M or Q2) - 50 th percentile
• Third/upper quartile (Q3) - 75 th percentile

You will use percentiles as you progress through your statistics and probability units, along with standard deviation (a measure of dispersion or central tendency), determining outliers, using the normal distribution, including quantiles and deciles, and more exploratory data analysis techniques.

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Statistics and Probability (6.SP.B.4) Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

How to find a quartile for a small data set

In order to find the quartiles for a small data set:

Order the data and find the median \textbf{(Q2).}

Count the number of data items in the set. Highlight the median and find the halfway point in the lower half of the data \textbf{(Q1).}

Highlight the median and find the halfway point in the upper half of the data \textbf{(Q3).}

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Quartile examples

Example 1: small discrete data set, odd number of data points.

Find the lower and upper quartiles.

The data is ordered, so you can go straight ahead and find the median. For a small data set, you can just cross numbers off from either end one by one until you reach the middle.

The median (Q2) is 7.

2 Count the number of data items in the set. Highlight the median and find the halfway point in the lower half of the data \textbf{(Q1).}

There are 11 items of data. As there is an odd number of data items, you do not include the median when looking at the lower half of the data.

If you mark 7 (the median) using two vertical lines either side of the number, as if to fence off the value from further calculations, the lower half of the data is

Repeating the process of crossing off to find the middle value for the lower half of the data, you have

The lower quartile (Q1) is 2.

3 Highlight the median and find the halfway point in the upper half of the data \textbf{(Q3).}

Excluding the median, the upper half of the data is

Repeating the process of crossing off to find the middle value for the upper half of the data, you have

The upper quartile (Q3) is 10.

Example 2: small continuous data set, even number of data points

The data below shows birth weights of 10 babies in kilograms.

Ordering the data, you get

Now, find the median by crossing numbers off from either end one by one until you reach the middle.

There are two middle values, so you need to calculate the midpoint of these.

(3.3+3.5)\div{2}=3.4

The median (Q2) is 3.4{~kg}.

The median was 3.4. Marking this on our set of data, you still do not include this value when calculating the lower quartile, but you do find the middle of all of the values below the median.

You mark 3.4, which is the median. The lower half of the data is

You do include the 3.3 value you used in the median calculation, as this wasn’t actually the median value.

Repeating the process of crossing off values to find the middle value for the lower half of the data, you have

The lower quartile (Q1) is 3.0{~kg}.

Again, note that you do include the data point 3.5.

Repeat the process of crossing off to find the middle value for the upper half of the data.

The upper quartile (Q3) is 3.7{~kg}.

Example 3: small discrete data set, using midpoints for quartiles

Find the lower and upper quartiles for the following data set.

Ordering this data set from smallest to largest, you have

As this is a small data set, just cross numbers off from either end in turn until you reach the middle.

The median (Q2) is 14.

As there is an odd number of data values, you do not use the median value when finding the other two quartiles. You mark 14, which is the median and so the lower half of the data is

As there are two data points, 5 and 8, you need to calculate the midpoint of these two values.

\cfrac{5+8}{2}=\cfrac{13}{2}=6.5

The lower quartile (Q1) is therefore 6.5.

The values in the upper half of the data are

Again, note that you don’t include the median data point in the upper half.

Again, there are two data points, 20 and 34, in the middle of the data, so you find the midpoint of these, which is 27.

If you weren’t sure on the midpoint, you could do the calculation

\cfrac{20+34}{2}=\cfrac{54}{2}=27

The upper quartile (Q3) is 27.

Working with a large data set

For a large data set, crossing numbers off a list can be time-consuming and a bit confusing, particularly if the data spans over two or more lines when listed. You can use an alternative method to find the lower and upper quartiles.

For a data set containing n values:

• The lower quartile , Q1, is located at the position \cfrac{n+1}{4}.
• The median , M or Q2, is located at the position \cfrac{n+1}{2}.
• The upper quartile , Q3, is located at the position 3\times\cfrac{n+1}{4}.

Note that you are finding out the position of the lower or upper quartile. You still need to count through the data set to find which values these are .

In situations where data is grouped, this method can also be used to find the class intervals in which the lower and upper quartiles lie. This is particularly true when estimating the quartiles in a histogram.

Note: The quartile function in Google Sheets or Microsoft Excel can help determine quartiles for very large sets of data.

How to find quartiles for a large data set

In order to find the lower quartile and upper quartile for a large data set:

Order the data and find the value of \textbf{n} (the number of data points).

Use the formula \bf{\cfrac{\textbf{n}+1}{4}} to calculate the position of \textbf{Q1} and state/calculate the data value at this position.

Use the formula \bf{3\times\cfrac{\textbf{n}+1}{4}} to calculate the position of \textbf{Q3} and state/calculate the data value at this position.

Quartile for a large data set examples

Example 4: large data set.

A set of data has been arranged into a stem and leaf plot. Calculate the value of each quartile for the following data.

The data is already ordered. There are 24 data points, so n=24.

Using the formula \cfrac{n+1}{4} with n=24, you have

\cfrac{24+1}{4}=\cfrac{25}{4}=6.25

So the lower quartile lies between the 6 th and 7 th data point.

The values are 5 and 6, and the midpoint of these is 5.5, so the lower quartile is 5.5.

You use the formula 3\times\cfrac{n+1}{4} with n=24 to give

3 \times \cfrac{24+1}{4}=3 \times \cfrac{25}{4}=3 \times 6.25=18.75

Note that if you’ve already found that the lower quartile is the 6.25 th value, you can simply multiply 6.25 by 3 to get 18.75.

So the upper quartile lies between the 18 th and 19 th data points, and you use the midpoint of these values.

The values are 19 and 20, and the midpoint of these is 19.5, so the upper quartile is 19.5.

Example 5: grouped data

The heights (h) of 40 students are given in the table below. In which class intervals do the lower and upper quartiles lie?

You don’t need to order the data. There are 40 data points, so n=40.

Note, this is given information in the actual question.

Using the formula \cfrac{n+1}{4} with n=40, you have

\cfrac{40+1}{4}=\cfrac{41}{4}=10.25

So the lower quartile value lies between the 10 th and 11 th data points.

The 10 th and the 11 th values both lie in the class interval 1.5 < h \leq {1.6} because, by using the cumulative frequency, you can see that 15 items of data have a height h\leq{1.6}\text{m}.

The lower quartile lies in the group 1.5<h\leq{1.6}.

Using the formula 3\times\cfrac{n+1}{4} with n=40, you have

3 \times \cfrac{40+1}{4}=3 \times \cfrac{41}{4}=3 \times 10.25=30.75

So the upper quartile value lies between the 30 th and 31 st data points.

The 30 th and the 31 st values both lie in the class interval 1.7 < h \leq {1.8} because, by using the cumulative frequency, you can see that 28 items of data have a height h\leq{1.7}\text{m}.

The upper quartile lies in the group 1.7<h\leq{1.8}.

Example 6: quartiles from a cumulative frequency diagram

The cumulative frequency diagram below shows the distribution of building heights in a city, in meters.

Use the diagram to calculate the values for Q1 and Q3 for this set of data.

The data within a cumulative frequency diagram is already ordered from smallest to largest. The value n is the highest value for the cumulative frequency on the graph. Here, n=80.

Using \cfrac{n+1}{4} with n=80, the position of Q1 is

\cfrac{80+1}{4}=\cfrac{81}{4}=20.25

This is the value on the cumulative frequency axis. Drawing a horizontal line to the curve, and then down to the x -axis, you can see that the value for the lower quartile is approximately 30{~m} (see diagram below).

The upper quartile is located at 3\times\cfrac{n+1}{4}. When n=20

Q_{1}=3\times\cfrac{20+1}{4}=3\times\cfrac{21}{4}=3\times{20.25}=60.75

Drawing a horizontal line from the cumulative frequency axis at the value 60.25, across to the curve and then down to the x -axis, the upper quartile is approximately 42{~m} (see diagram below).

Teaching tips for quartile

• Use visual aids such as number lines, bar graphs, or dot plots to illustrate quartiles. Show students how to locate Q1, Q2, and Q3 for any given set of values. Encourage them to interpret and analyze the data based on the quartiles.
• Make the concept relatable by using real-life examples that students can connect with. For instance, use numeric values related to their favorite sports, hobbies, or interests. This helps students see the relevance of quartiles in analyzing and understanding data.
• Provide ample opportunities for students to practice finding quartiles using different data sets. Offer exercises with varying levels of difficulty to challenge and reinforce their understanding.
• Utilize interactive tools or online resources that allow for manipulating, exploring, and even calculating quartiles to manipulate and explore quartiles. For example, use virtual manipulatives or educational websites that provide interactive activities related to quartiles.

Easy mistakes to make

• Forgetting to order the data set before finding the median or quartiles The list must be in order before you start finding the key values!
• Stating the quartile value as the number given by the formula, rather than counting and finding that value in the data For example, for the data set 3, \, 4, \, 6, \, 7, \, 10, the formula for the lower quartile gives \cfrac{5+1}{4}=\cfrac{6}{4}=1.5. This tells you which data values to select (in this case, the midpoint of the 1 st and 2 nd values); do not just write Q1=1.5.

Related representing data lessons

• Representing data
• Stem and leaf plot

Practice quartile questions

1) Find the values of the lower and upper quartiles for the following 11 items of data.

The median is 17, so mark this, then find the median of the lower half of the data (4, \, 6, \, 6, \, 7, \, 12) and the median of the upper half of the data (18, \, 20, \, 21, \, 32,3 \, 5).

Q1 is 6 and Q2 is 21.

2) Find the lower and upper quartiles.

The median is the middle number in an ordered list. The list is already ordered, but there are two values in the middle of the data.

As they are both 13, the median is 13.

The middle of the lower half of the data (2, \, 4, \, 7, \, 10, \, 13) is 7 so the lower quartile is 7.

The middle of the upper half of the data (13, \, 15, \, 19, \, 22, \, 32) is 19 so the upper quartile is 19.

3) Find the lower and upper quartiles for the following set of data.

Ordering the data first, you have

The median is the midpoint of 0 and 4; mark this, then find the median of the lower half of the data (-3, \, -2, \, -2, \, 0) and the median of the upper half of the data (4, \, 5, \, 10, \, 19).

There are an even number of data points, so use the midpoint of the 2 nd and 3 rd for Q1 and the midpoint of the 6 th and 7 th for Q3.

4) This table shows the shoe sizes of 15 girls. Find the lower and upper quartiles.

Use the formula to find out which values to pick for the lower quartile and the upper quartile. There are 15 pieces of data, so n=15.

Q1=\cfrac{15+1}{4}=4 so the lower quartile is the 4 th data point, which is 5.

Q3=3\times\cfrac{15+1}{4}=12 so the upper quartile is the 12 th data point, which is 7.

5) Raheem writes down how long his bus journey to school takes on 20 different days. The times are given to the nearest minute. Find the lower and upper quartiles for his data set.

Use the formula to find out which values to pick for the lower quartile and the upper quartile. There are 20 data points, so n=20.

Q1=\cfrac{20+1}{4}=5.25 so by finding the midpoint of the 5 th and 6 th data points, which are both 11 (remember to use the key), the lower quartile is 11.

Q3=3\times\cfrac{15+1}{4}=15.75 so the midpoint of the 15 th and 16 th data points, which are both 18 (using the key), gives us the upper quartile of 18.

6) The weights (w) of 30 teenagers are given in the table below. Which class interval contains the upper quartile?

Using the formula 3\times\cfrac{n+1}{4} to determine the upper quartile with n=1+8+10+7+4=30, you have

Q3=3\times\cfrac{30+1}{4}=3\times\cfrac{31}{4}=23.25

So the upper quartile is between the 23 rd and 24 th data points.

Counting the cumulative frequency for the data set, you have

The 23 rd and 24 th data points both lie in the class interval 60<w\leq{70}.

Quartile FAQs

A quartile is a type of quantile that divides an ordered data set into four equal parts (quarters).

To find each quartile of a data set, first sort the data set in ascending order from smallest to largest and find the median (Q2). Find the lower quartile (Q1) by locating the median (central value) of the lower half of the data set, and find the upper quartile (Q3) by locating the median (central value) of the upper half of the data set.

The next lessons are

• Units of measurement
• Represent and interpret data
• Frequency graph

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• CBSE Class 11 Statistics for Economics Notes

Chapter 1: Concept of Economics and Significance of Statistics in Economics

• Statistics for Economics | Functions, Importance, and Limitations

Chapter 2: Collection of Data

• What is Data Collection? | Methods of Collecting Data
• Sources of Data Collection | Primary and Secondary Sources
• Direct Personal Investigation: Meaning, Suitability, Merits, Demerits and Precautions
• Indirect Oral Investigation: Meaning, Suitability, Merits, Demerits and Precautions
• Difference between Direct Personal Investigation and Indirect Oral Investigation
• Information from Local Source or Correspondents: Meaning, Suitability, Merits, and Demerits
• Questionnaires and Schedules Method of Data Collection
• Difference between Questionnaire and Schedule
• Qualities of a Good Questionnaire and types of Questions
• What are the Published Sources of Collecting Secondary Data?
• What Precautions should be taken before using Secondary Data?
• Two Important Sources of Secondary Data: Census of India and Reports & Publications of NSSO
• What is National Sample Survey Organisation (NSSO)?
• What is Census Method of Collecting Data?
• Sample Method of Collection of Data
• Methods of Sampling
• Father of Indian Census
• What makes a Sampling Data Reliable?
• Difference between Census Method and Sampling Method of Collecting Data
• What are Statistical Errors?

Chapter 3: Organisation of Data

• Organization of Data
• Objectives and Characteristics of Classification of Data
• Classification of Data in Statistics | Meaning and Basis of Classification of Data
• Concept of Variable and Raw Data
• Types of Statistical Series
• Difference between Frequency Array and Frequency Distribution
• Types of Frequency Distribution

Chapter 4: Presentation of Data: Textual and Tabular

• Textual Presentation of Data: Meaning, Suitability, and Drawbacks
• Tabular Presentation of Data: Meaning, Objectives, Features and Merits
• Different Types of Tables
• Classification and Tabulation of Data

Chapter 5: Diagrammatic Presentation of Data

• Diagrammatic Presentation of Data: Meaning , Features, Guidelines, Advantages and Disadvantages
• Types of Diagrams
• Bar Graph | Meaning, Types, and Examples
• Pie Diagrams | Meaning, Example and Steps to Construct
• Histogram | Meaning, Example, Types and Steps to Draw
• Frequency Polygon | Meaning, Steps to Draw and Examples
• Ogive (Cumulative Frequency Curve) and its Types
• What is Arithmetic Line-Graph or Time-Series Graph?
• Diagrammatic and Graphic Presentation of Data

Chapter 6: Measures of Central Tendency: Arithmetic Mean

• Measures of Central Tendency in Statistics
• Arithmetic Mean: Meaning, Example, Types, Merits, and Demerits
• What is Simple Arithmetic Mean?
• Calculation of Mean in Individual Series | Formula of Mean
• Calculation of Mean in Discrete Series | Formula of Mean
• Calculation of Mean in Continuous Series | Formula of Mean
• Calculation of Arithmetic Mean in Special Cases
• Weighted Arithmetic Mean

Chapter 7: Measures of Central Tendency: Median and Mode

• Median(Measures of Central Tendency): Meaning, Formula, Merits, Demerits, and Examples
• Calculation of Median for Different Types of Statistical Series
• Calculation of Median in Individual Series | Formula of Median
• Calculation of Median in Discrete Series | Formula of Median
• Calculation of Median in Continuous Series | Formula of Median
• Graphical determination of Median
• Mode: Meaning, Formula, Merits, Demerits, and Examples
• Calculation of Mode in Individual Series | Formula of Mode
• Calculation of Mode in Discrete Series | Formula of Mode
• Grouping Method of Calculating Mode in Discrete Series | Formula of Mode
• Calculation of Mode in Continuous Series | Formula of Mode
• Calculation of Mode in Special Cases
• Calculation of Mode by Graphical Method
• Mean, Median and Mode| Comparison, Relationship and Calculation

Chapter 8: Measures of Dispersion

• Measures of Dispersion | Meaning, Absolute and Relative Measures of Dispersion
• Range | Meaning, Coefficient of Range, Merits and Demerits, Calculation of Range
• Calculation of Range and Coefficient of Range
• Interquartile Range and Quartile Deviation

Partition Value | Quartiles, Deciles and Percentiles

• Quartile Deviation and Coefficient of Quartile Deviation: Meaning, Formula, Calculation, and Examples
• Calculation of Mean Deviation for different types of Statistical Series
• Mean Deviation from Mean | Individual, Discrete, and Continuous Series
• Standard Deviation: Meaning, Coefficient of Standard Deviation, Merits, and Demerits
• Standard Deviation in Individual Series
• Methods of Calculating Standard Deviation in Discrete Series
• Methods of calculation of Standard Deviation in frequency distribution series
• Combined Standard Deviation: Meaning, Formula, and Example
• How to calculate Variance?
• Coefficient of Variation: Meaning, Formula and Examples
• Lorenz Curve: Meaning, Construction, and Application

Chapter 9: Correlation

• Correlation: Meaning, Significance, Types and Degree of Correlation
• Methods of measurements of Correlation
• Calculation of Correlation with Scattered Diagram
• Spearman's Rank Correlation Coefficient
• Karl Pearson's Coefficient of Correlation
• Karl Pearson's Coefficient of Correlation | Methods and Examples

Chapter 10: Index Number

• Index Number | Meaning, Characteristics, Uses and Limitations
• Methods of Construction of Index Number
• Unweighted or Simple Index Numbers: Meaning and Methods
• Methods of calculating Weighted Index Numbers
• Fisher's Index Number as an Ideal Method
• Fisher's Method of calculating Weighted Index Number
• Paasche's Method of calculating Weighted Index Number
• Laspeyre's Method of calculating Weighted Index Number
• Laspeyre's, Paasche's, and Fisher's Methods of Calculating Index Number
• Consumer Price Index (CPI) or Cost of Living Index Number: Construction of Consumer Price Index|Difficulties and Uses of Consumer Price Index
• Methods of Constructing Consumer Price Index (CPI)
• Wholesale Price Index (WPI) | Meaning, Uses, Merits, and Demerits
• Index Number of Industrial Production: Meaning, Characteristics, Construction, and Example
• Inflation and Index Number

Important Formulas in Statistics for Economics

• Important Formulas in Statistics for Economics | Class 11

What do you mean by Partition Value?

Quartile, Decile, and Percentiles of partition values represent various perspectives on the same subject. To put it another way, these are values that partition the same collection of observations in several ways. As a result, it can divide these into many equal parts. According to the definition of the median, it is the middle point in the axis frequency distribution curve, and it divides the area under the curve into two areas with the same area on the left and right. The area under the curve for four equally divided parts of the area is called quartiles , the area for ten equally divided parts of the area is called deciles , and the area for hundred equally divided parts of the area is named percentiles .

(I) Quartiles

There are several ways to divide an observation when required. To divide the observation into two equally-sized parts, the median can be used. A quartile is a set of values that divides a dataset into four equal parts. The first quartile, second quartile, and third quartile are the three basic quartile categories. The lower quartile is another name for the first quartile and is denoted by the letter Q 1 . The median is another term for the second quartile and is denoted by the letter Q 2 . The third quartile is often referred to as the upper quartile and is denoted by the letter Q 3 .

Simply put, the values which divide a list of numerical data into three quarters are known as quartiles . The center of the three quadrants measures the central point of distribution and displays data that are close to the center. The quartiles represent the distribution or dispersion of the data set as a whole. The formulas for calculating quartiles are:

where, n is the total number of observations, Q 1 is First Quartile, Q 2 is Second Quartile, and Q 3 is Third Quartile.

Example 1: 

Calculate the lower and upper quartiles of the following weights in the family: 25, 17, 32, 11, 40, 35, 13, 5, and 46.

Solution: 

First of all, organise the numbers in ascending order.

5, 11, 13, 17, 25, 32, 35, 40, 46

Q 1 = 2.5 th term

As per the quartile formula;

Q 1 = 2 nd term + 0.5(3 rd term – 2 nd term)

Q 1 = 11 + 0.5(13 – 11) = 12

Q 3 = 7.5 th item

Q 3 = 7 th term + 0.5(8 th term – 7 th term)

Q 3 = 35 + 0.5(40 – 35) = 37.5

Calculate Q1 and Q3 for the data related to the age in years of 99 members in a housing society.

Q 1 = 25 th item

Now, the 25 th item falls under the cumulative frequency of 25 and the age against this cf value is 18.

Q 1 = 18 years

Q 3 = 75 th item

Now, the 75 th item falls under the cumulative frequency of 85 and the age against this cf value is 40.

Q 3 = 40 years

Example 3: 

Determine the quartiles Q 1 and Q 3 for the company’s salary listed below.

= 15 th item

Now, the 15 th item falls under the cumulative frequency 22 and the salary against this cf value lies in the group 600-700.

Q 1 = ₹641.67

Q 3 = 45 th item

Now, the 45 th item falls under the cumulative frequency 52 and the salary against this cf value lies in the group 800-900.

(II) Deciles

The deciles involve dividing a dataset into ten equal parts based on numerical values. There are therefore nine deciles altogether. Deciles are represented as follows: D 1 , D 2 , D 3 , D 4 ,…………, D 9 . One-tenth (1/10) of every given observation must be less than or equal to D 1 , for D 1 to be the usual highest value. The remaining nine-tenths of the same observation, or 9/10, is, nevertheless, greater than or equal to D 1 ‘s value.

A decile is used to group big data sets in descriptive statistics either from highest to lowest values or vice versa. It takes place on a scale from one to ten, with each number after that representing a rise of ten percentage points. A decile is a type of quantile that, like the quartile and the percentile, divides a set of data into groups that are simple to measure and analyse. In the domains of finance and economics, this kind of data ranking is carried out as a part of several academic and statistical studies. The formulas for calculating deciles are:

………… Where, n is the total number of observations, D 1 is First Decile, D 2 is Second Decile,……….D 9 is Ninth Decile.

Calculate the D 1 , D 5 from the following weights in a family: 25, 17, 32, 11, 40, 35, 13, 5, and 46.

D 1 = 1 st item = 5

D 5 = 5 th item = 25

Example 2: 

Calculate D 2 and D 6 for the data related to the age (in years) of 99 members in a housing society.

D 2 = 20 th item

Now, the 20 th item falls under the cumulative frequency of 25 and the age against this cf value is 18.

D 2 = 18 years

D 6 = 60 th item

Now, the 60 th item falls under the cumulative frequency of 65 and the age against this cf value is 35.

D 6 = 35 years

Determine D 4 for the company’s salary listed below.

In case N is an even number, the following formula is used:

D 4 = 24 th item

Now, the 24 th item falls under the cumulative frequency 22 and the salary against this cf value lies in the group 700-800.

D 4 =  ₹712.5

(III) Percentiles

Centiles is another term for percentiles. Any given observation is essentially divided into a total of 100 equal parts by a centile or percentile. These percentiles or centiles are represented as P 1 , P 2 , P 3 , P 4 ,……….P 99 . P 1 is a typical value of peaks for which 1/100 of any given data is either less than P 1 or equal to P 1 .

Usually, the percentile formula is used to compare a person’s performance to that of others. Remembering that a candidate’s percentile in tests and scores indicates how they rank among other candidates is important. The ratio of values below the value ‘x’ to all matters is used to determine the percentile of the value ‘x’. The percentiles can be calculated for weight, wealth, and several other factors. The formulas for calculating percentiles are:

……………. Where, n is the total number of observations, P 1 is First Percentile, P 2 is Second Percentile, P 3 is Third Percentile, ……….P 99 is Ninety Ninth Percentile.

Calculate the P 20 , P 90 from the following weights in the family: 25, 17, 32, 11, 40, 35, 13, 5, and 46.

Solution:  

P 20 = 2 nd item

P 90 = 9 th item

Calculate P 10 and P 75 for the data related to the age (in years) of 99 members in a housing society.

P 10 = 10 th item

Now, the 10 th item falls under the cumulative frequency of 20 and the age against this cf value is 10.

P 10 = 10 years

P 75 = 75 th item

P 75 = 40 years

Determine the value of P 50 for the company’s salary listed below.

P 50 = 30 th item

Now, the 30 th item falls under the cumulative frequency 38 and the salary against this cf value lies between 700-800.

P 50 = ₹750

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How to Determine Measures of Position (Percentiles and Quartiles)

Although you may not often use measures such as percentiles and quartiles, these values are used to describe data in some situations, and knowing how to interpret them is beneficial.

o       Range

o       Measure of position

o       Percentile

o       Quartile

o       Determine the range of a data set

o       Know how to interpret and determine measures of position (percentiles and quartiles)

While measures of central tendency, dispersion, and skewness are used often in statistics, there are other methods of characterizing or describing data distributions or portions that are commonly used as well. We will examine several of these statistical measures, some of which you may already know or have seen elsewhere.

The range of a data set is simply the difference between the maximum and minimum values of the set. (This measure is typically considered a measure of dispersion, since it is a simple description of how far the data extends.) Thus, if a data set such as { x 1 , x 2 , x 3 ,., x N } is provided in increasing order so that x i < x i +1 , then the range of the data set is simply x N  – x 1 . If the data set is not ordered, then you must simply determine by inspection the maximum and minimum values.

Practice Problem: Find the range of the following data set.

{1, 6, 3, 9, 7, 2, 5, 4, 11, 15, 10}

Solution : We can find the range either by simply looking for the maximum and minimum values or by arranging the set in increasing order and then subtracting the first element from the last. Although the latter approach is a bit more time consuming, it can be beneficial in cases where you need to perform other calculations. So, let's order the data set for the sake of completeness.

{1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 15}

The range is then 15 – 1 = 14.

Quartiles and Percentiles

Virtually anyone who has taken a standardized test at one time or another is familiar with the term percentile . Although percentiles seem dangerously similar to percentages (that is, "percent correct," referring to the number of questions answered correctly divided by the total number of questions, all multiplied by 100%), they are actually different. A similar measurement is the quartile, which we will also discuss. Both percentiles and quartiles are statistical measures of position ; that is, they do not measure a central tendency or a spread (dispersion), but instead measure location in a data set. (The exact definition of a percentile and quartile differs; these differences, however, tend to be minor and are focused on certain fine points. Also, these differences tend to disappear when the number of data values in the set is large.)

Let's consider a number p , where p is a whole number between 0 and 100. Assume that the number p describes the percentage of values less than or equal to some data value N p . Consequently, 100 – p is the percentage of values greater than N p . This number N p is the p th percentile . Thus, to say that some data value x is the 75th percentile is to say that 75% of all the values in the data set are less than or equal to x , and that 25% of the data values are greater than x . Note that the percentile of a data value can also be understood as 100 times the cumulative relative frequency of that value. (Recall that the cumulative relative frequency of a value x is the relative frequency of all values less than or equal to x .) So, a student who gets a test score in the 90th percentile, for instance, hasn't (necessarily) scored 90/100 correct--he simply has a score that is at least as good as 90% of the other students. Although such a description isn't necessarily very satisfying for the student (who is probably more interested in finding out his percentage of correct answers), it is statistically helpful in certain situations. Typically, the 0th and 100th percentiles are not discussed, because these values are simply the minimum and maximum (respectively) of the data set.

Practice Problem: For the data set below, which value is in the 75th percentile?

{1, 3, 3, 4, 6, 7, 7, 7, 8, 9, 9, 10, 12, 15, 16, 17}

Solution : We want to find the data value N p for which 75% of the data set is less than or equal to N p . Note that there are a total of 16 values in the set; thus, 75% of the data set is 12 values. Because the data set is ordered, we need simply find the 12th data value; then, 75% (12 out of 16 values) of the data set will be less than or equal to this value. The number 10 is the 75th percentile: 75% of the values in the set are less than or equal to 10.

Practice Problem: Which of the following data values is the 50th percentile?

{1.52, 5.36, 6.79, 5.21, 0.28, 6.36, 8.47, 5.52, 6.26, 5.97}

Solution : The 50th percentile is that value N for which 50% of the values in the set are less than or equal to N . To help us find this value, let's first order the data set.

{0.28, 1.52, 5.21, 5.36, 5.52, 5.97, 6.26, 6.36, 6.79, 8.47}

The data set has 10 values; thus, the 50th percentile is the fifth data value, 5.52. Exactly half (50%) of the data values are less than or equal to 5.52, and the remaining half are greater than 5.52.

Another measure of position is the quartile , which is similar to the percentile except that it divides data into quarters (segments of 25% each) instead of hundredths. Thus, the n th quartile is the value x for which (25 n )% of the values are less than or equal to x . Three quartiles are defined: Q1, Q2, and Q3. The quartile Q1 corresponds to the 25th percentile, Q2 to the 50th percentile, and Q3 to the 75th percentile.

The Q2 and the 50th percentile are sometimes said to correspond to the median of a data set. Given our definition of a median, this is true when there are an odd number of data values; it is not strictly true for an even number of data values (see the practice problem above)--the median, according to our definition, would actually be the mean of 5.52 and 5.97. We could, however, say that this median value (5.75) is the 50th percentile for the data set: technically, half the values in the data set are below this value, and half are above. Thus, we can still maintain our definition of the median if we appropriately define percentiles and quartiles. In addition, we can also note that Q1 is the median of the first half of the values, and Q3 is the median of the second half of the values. (Our above considerations on the definition of the median apply here as well.)

Practice Problem : What is Q3 for the following data set?

{20, 40, 50, 65, 70, 75, 80, 100}

Solution : Q3 is the value x for which 75% (three out of four) of the data values are at most x . Since there are eight members in the data set, the sixth value is Q3-75. This value is also the 75th percentile.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 4.

• Calculating percentile

Calculate percentiles

• Analyzing a cumulative relative frequency graph
• Cumulative relative frequency graph problem
• (Choice A)   58 th ‍   percentile A 58 th ‍   percentile
• (Choice B)   67 th ‍   percentile B 67 th ‍   percentile
• (Choice C)   75 th ‍   percentile C 75 th ‍   percentile
• (Choice D)   83 th ‍   percentile D 83 th ‍   percentile