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

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

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. favourite colour)

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 fraction numerator n plus 1 over denominator 2 end fraction 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

Find the median of the data set given below.

43                        29                        70                        51                        64                       43

Answer:

First put the values in ascending size order

29                        43                        43                        51                        64                       70

Find the position of the middle value, using fraction numerator n plus 1 over denominator 2 end fraction

fraction numerator 6 plus 1 over denominator 2 end fraction equals 7 over 2 equals 3.5

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

Calculate the median

fraction numerator 43 plus 51 over denominator 2 end fraction equals 47

The median value is 47

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

    • x with bar on top equals 1 over n sum from i equals 1 to n of x subscript i equals fraction numerator sum from i equals 1 to n of x subscript i over denominator n end fraction to the power of blank

    • Where sum from i equals 1 to n of x subscript i equals x subscript 1 plus x subscript 2 plus... plus x subscript n is the sum of the n pieces of data

Worked Example

Find the mean of the data set given below.

 27                        31                        9                        65                        52                       43

Answer:

Use the formula to calculate the mean

fraction numerator 27 plus 31 plus 9 plus 65 plus 52 plus 43 over denominator 6 end fraction equals 37.83333...

The mean is 37.8

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Naomi C

Author: Naomi C

Expertise: Maths Content Creator

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

Reviewer: 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.

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