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AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable DataSummary StatisticsGrouped Data

Grouped Data (College Board AP® Statistics)

Study Guide

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Mean from grouped data

How do I find the mean for grouped data?

  • The exact mean from a set of grouped data cannot be found

    • but an estimate can be calculated

  • To find an estimate for the mean of grouped data in a table:

    • Create two new columns for the table and label them x and f x

    • Find the midpoint of each group and write it in the column labeled x

    • Multiply the frequency of each group by the group's midpoint and write it in the column labeled f x

    • Find the sum of these values

    • Divide the sum by the total frequency

  • The formula to find the mean of grouped data is x with bar on top equals fraction numerator sum for blank of f x over denominator sum for blank of f end fraction

    • This is not given to you in the exam

Worked Example

The weights of 20 three-week-old Labrador puppies were recorded at a vet's clinic. The results are shown in the table below.

Weight, w kg

Frequency

3 ≤ w < 3.5

3

3.5 ≤ w < 4

4

4 ≤ w < 4.5

6

4.5 ≤ w < 5

5

5 ≤ w < 6

2

Estimate the mean weight of these puppies.

Answer:

Add two columns to the table and complete the first new column with the midpoints of the class intervals

Complete the second extra column by calculating f x

It is worth also adding a total row at the bottom of the table

Weight, w kg

Frequency

Midpoint

f x

3 ≤ w < 3.5

3

3.25

3 × 3.25 = 9.75

3.5 ≤ w < 4

4

3.75

4 × 3.75 = 15

4 ≤ w < 4.5

6

4.25

6 × 4.25 = 25.5

4.5 ≤ w < 5

5

4.75

5 × 4.75 = 23.75

5 ≤ w < 6

2

5.5

2 × 5.5 = 11

Total

20

85

Find the estimate of the mean by dividing the the total of the f x column by the total of the frequency column, fraction numerator sum for blank of f x over denominator sum for blank of f end fraction

Mean equals 85 over 20 equals 4.25

An estimate of the mean weight of the puppies is 4.25 kg

Median from grouped data

How do I find the median for grouped data?

  • The exact median from a set of grouped data cannot be found

    • but the group that contains the median can be identified

  • The position of the median can be found using fraction numerator n plus 1 over denominator 2 end fraction

    • where n is the total number of data values (total frequency)

  • For grouped data in a table

    • Find the interval containing the open parentheses fraction numerator n plus 1 over denominator 2 end fraction close parentheses to the power of th value (median)

    • You could add another column to the table containing the cumulative frequency to help you find this

      • This is a column that adds up the frequency as you go along

Worked Example

The heights of 30 mature oak trees were measured and recorded in the table below.

Height, h m

Frequency

22 ≤ h < 23

4

23 ≤ h < 24

7

24 ≤ h < 25

6

25 ≤ h < 26

8

26 ≤ h < 27

3

27 ≤ h < 28

2

Find the height interval that contains the median height of the mature oak trees.

Answer:

Calculate the position of the median within the data set

fraction numerator 30 plus 1 over denominator 2 end fraction equals 15.5

This means that the median lies between the 15th and 16th data values

Add a 'cumulative frequency' column to the table to add up the frequency as you go along

Height, h m

Frequency

Cumulative frequency

22 ≤ h < 23

4

4

23 ≤ h < 24

7

11

24 ≤ h < 25

6

17

25 ≤ h < 26

8

25

26 ≤ h < 27

3

28

27 ≤ h < 28

2

30

Identify the group(s) that contain the 15th and 16th values

Both the 15th and 16th values are in the group 24 ≤ h < 25

Therefore the median must also lie in this group

The median height of the mature oak trees lies in the interval 24 ≤ h < 25, where h is the height in meters

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.