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1. Data Distributions
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VCE 11 General 2023

1.04 Measures of centre

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Introduction

It is often very useful to determine the centre of a data distribution to be able to compare the data to other sets, or to summarise key features of the data. Three measures of centre are the mean (average), median and the mode.

Mean

The mean is often referred to as the average. To calculate the mean, add all the scores in a data set, then divide this by number of scores.

The mean of a data set, denoted \overline{x}, is given by \overline{x}=\dfrac{\Sigma x}{n}.

Examples

Example 1

The mean of 4 scores is 21. If three of the scores are 17,\,3, and 8, find the 4th score, x.

Worked Solution
Create a strategy

Use the formula: \overline{x}=\dfrac{\Sigma x}{n}

Apply the idea

The given values are \overline{x}=21,\, \Sigma x =17+3+8+x, and n=4.

\displaystyle 21\displaystyle =\displaystyle \dfrac{17+3+8+x}{4}Substitute the values
\displaystyle 21\displaystyle =\displaystyle \dfrac{28+x}{4}Perform the addition
\displaystyle 84\displaystyle =\displaystyle 28+xMultiply both sides by 4
\displaystyle x\displaystyle =\displaystyle 56Subtract 28 from both sides
Idea summary

The mean of a data set, denoted \overline{x}, is given by \overline{x}=\dfrac{\Sigma x}{n}

Median

The median is the middle score in a data set, when the data is sorted to be in order. If there are two middle scores, we take the average of them to find the median.

To find the median of a data set:

  • Sort the data points into order

  • If there are an odd number of scores, the median is the middle score (the \dfrac{n+1}{2} score

  • If there are an even number of scores, the median is calculated by finding the mean of the two middle scores (the \dfrac{n}{2}th and \left(\dfrac{n}{2} + 1\right) th scores.)

Examples

Example 2

Find the median of the nine numbers below:1,\,1,\,3,\,5,\,7,\,9,\,9,\,10,\,15

Worked Solution
Create a strategy

The median in an odd set of scores is the \left( \dfrac{n+1}{2} \right)th score, where n is the total number of scores.

Apply the idea

There are 9 scores.

\displaystyle \text{Median position}\displaystyle =\displaystyle \dfrac{9+1}{2} Substitute the values
\displaystyle =\displaystyle \dfrac{10}{2} Evaluate the numerator
\displaystyle =\displaystyle 5\text{th score} Perform the division

Therefore the median will be the 5th score. \text{Median}=7

Reflect and check

This data set had an odd number of scores, so the median was just the middle score.

Example 3

Find the median from the frequency distribution table:

\text{Score }(x)\text{Frequency }(f)
232
2426
2537
2624
2725
Worked Solution
Create a strategy

To find the median we need to add a cumulative frequency column.

Apply the idea

To find the median, we create a cumulative frequency column.

ScoreFrequencyCumulative frequency
2322
24262+26=28
253728+37=65
262465+24=89
272589+25=114

Since there are 114 scores, the median will be the average of the 57th and 58th score. Looking at the cumulative frequency table, there are 28 scores less than or equal to 24 and 65 scores less than or equal to 25. This means that the 57th and 58th scores are both 25, so the median is 25.

\text{Median}=25

Idea summary

To find the median of a data set:

  • Sort the data points into order

  • If there are an odd number of scores, the median is the middle score (the \dfrac{n+1}{2} score

  • If there are an even number of scores, the median is calculated by finding the mean of the two middle scores (the \dfrac{n}{2}th and \left(\dfrac{n}{2} + 1\right) th scores.)

Mode

The mode describes the most frequently occurring score.

Suppose that 10 people were asked how many pets they had. 2 people said they didn't own any pets, 6 people had one pet and 2 people said they had two pets.

In this data set, the most common number of pets that people have is one pet, and so the mode of this data set is 1.

A data set can have more than one mode, if two or more scores are equally tied as the most frequently occurring.

The mode of a data set is the most frequently occurring score.

Examples

Example 4

Find the mode of the following scores:8,\,18,\,5,\,2,\,2,\,10,\,8,\,5,\,14,\,14,\,8,\,8,\,10,\,18,\,14,\,5

Worked Solution
Create a strategy

Choose which number in the set appeared the most.

Apply the idea

We can list the number and how many times it appears:

  • 8 appeared in the set 4 times.

  • 18 appeared in the set 2 times.

  • 5 appeared in the set 3 times.

  • 2 appeared in the set 2 times.

  • 10 appeared in the set 2 times.

  • 14 appeared in the set 3 times.

\text{Mode}= 8

Idea summary

The mode of a data set is the most frequently occurring score.

Outcomes

U1.AoS1.4

mean 𝑥 and sample standard deviation s

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