AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable Data and Collecting DataSummary StatisticsMeasures of Center

Measures of Center (College Board AP® Statistics): Study Guide

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 21 May 2026

Mode

What is the mode?

The mode is the value that occurs most often in a data set
- It is possible for there to be more than one mode
- It is possible for there to be no mode
The mode can appear at any point in the distribution of the data set
- It is not necessarily in the center of the distribution
The mode is the only measure of center that can be used for categorical data (e.g. favorite color)

Examiner Tips and Tricks

If there is no mode in a particular data set, state that there is no mode, do not say that the mode is zero!

Median

What is the median?

The median is the middle value when the data is in order of size
- If there are an odd number of values in the data set
  - then the median is the data value in the center of the set
- If there are an even number of values in the data set
  - then there are two values in the middle and the median is the midpoint of these two values

How do I find the median for ungrouped data?

For a data set that contains $n$ values
- the middle value will be located at the $\frac{n + 1}{2}$ position
The median is a resistant statistic
- It is not affected by extreme values
- It is a suitable measure of center for distributions that are
  - either approximately symmetrical
  - or strongly skewed with outliers

Worked Example

A wildlife biologist is studying a local population of box turtles. The biologist selects a random sample of 6 box turtles and records their weights (in ounces). The data are shown below:

43 29 70 51 64 43

Calculate the median weight of the turtles in the sample.

Answer:

First, put the values in ascending size order

29 43 43 51 64 70

Find the position of the middle value, using $\frac{n + 1}{2}$

$\frac{6 + 1}{2} = \frac{7}{2} = 3.5$

The middle value lies halfway between the 3rd and the 4th values

Calculate the median

$\frac{43 + 51}{2} = 47$

The median weight of the turtles is 47 ounces

Mean

What is the mean?

The mean is the sum of all the values divided by the number of values in the data set
- It uses all values in a data set
If the distribution of the data is approximately symmetrical
- the median and the mean will be close in value
  - and both values will be representative of the population
If the distribution of the data is strongly skewed or has outliers
- The mean will be affected by the extreme values
  - It is therefore not a resistant statistic

How do I find the mean for ungrouped data?

The formula for calculating the mean is given to you in the exam
- $\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$
- Where $\sum_{i = 1}^{n} x_{i} = x_{1} + x_{2} + . . . + x_{n}$ is the sum of the $n$ pieces of data

Worked Example

A botanist is studying the growth of a specific species of fern in a controlled greenhouse. The botanist selects a random sample of 6 ferns and records their heights (in centimeters) after one month of growth. The data are shown below:

27 31 9 65 52 43

(a) Calculate the mean height of the ferns in the sample. Interpret this value in context.

(b) The median height of this sample is 37 cm. Suppose the botanist discovers a 7th fern in the greenhouse that grew to a height of 140 cm and adds it to the dataset. Explain how adding this new observation will affect the mean and the median. Which measure of center should the botanist use to describe the typical height of the 7 ferns?

Answer:

(a)

Use the formula to calculate the mean

$\frac{27 + 31 + 9 + 65 + 52 + 43}{6} = 37.83333 . . .$

Interpret this in context

The average height for this sample of 6 ferns is 37.8 cm

(b)

Describe what would happen to the mean

Adding an extreme high value of 140 cm (an outlier) to the dataset will cause the mean to increase substantially

Describe what would happen to the median

The median will only shift slightly (to exactly 43 cm, the new middle value of the ordered list)

State which measure the botanist should use

The botanist should use the median because itis a resistant (robust) measure of center that is not heavily influenced by extreme values

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent

Measures of Center (College Board AP® Statistics): Study Guide

Mode

What is the mode?

Examiner Tips and Tricks

Median

What is the median?

How do I find the median for ungrouped data?

Worked Example

Mean

What is the mean?

How do I find the mean for ungrouped data?

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Naomi C

Reviewer: Dan Finlay