Grouped Data (College Board AP® Statistics)

Study Guide

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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 $\bar{x} = \frac{\sum_{} f x}{\sum_{} f}$
- 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, $\frac{\sum_{} f x}{\sum_{} f}$

Mean $= \frac{85}{20} = 4.25$

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 $\frac{n + 1}{2}$
- where $n$ is the total number of data values (total frequency)
For grouped data in a table
- Find the interval containing the ${(\frac{n + 1}{2})}^{th}$ 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

$\frac{30 + 1}{2} = 15.5$

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

Last updated: 16 September 2024

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Grouped Data (College Board AP® Statistics)

Study Guide

Mean from grouped data

How do I find the mean for grouped data?

Worked Example

Median from grouped data

How do I find the median for grouped data?

Worked Example

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Author: Naomi C

Author: Dan Finlay