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7.03 Mean and median as measures of center

Lesson

Sometimes we want to talk about a data set without having to refer to every single result. In other words, we want to summarize the data set to learn more about it and make comparisons.

In the last lesson, we introduced the mode, the most frequently occurring score. In this lesson, we will learn about two more ways we can summarize numerical data sets.

The mean

The mean of a data set is an average score.

Exploration

Three friends are planning a trip to Alice Springs. They plan to fly there, and discover that the airline imposes a weight limit on their luggage of $20$20 kg per person. On the night before the flight they weigh their luggage and find that their luggage weights form this data set:

$17,18,22$17,18,22

One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find:

$\frac{17+18+22}{3}=\frac{57}{3}=19$17+18+223=573=19

Each person carries $19$19 kg. This amount is the mean of the data set.

Summary

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same.

To calculate the mean, use the formula:

$\text{mean}=\frac{\text{sum of scores}}{\text{number of scores}}$mean=sum of scoresnumber of scores

Worked example

Find the mean of this data set:

$4,7,1,2,3$4,7,1,2,3

Think: There are $5$5 scores, so we should add these numbers all together and divide by $5$5.

Do: $4+7+1+2+3=17$4+7+1+2+3=17, and $17\div5=3.4$17÷​5=3.4.

Reflect: Even though all the numbers in the data set are whole numbers, the mean is a decimal. If the data set was produced from a survey question "How many siblings do you have?", we would say the mean number of siblings was $3.4$3.4, even though it isn't possible to have $0.4$0.4 siblings! The mean is a way to summarize data - it is not part of the data set itself.

 

The median

The median of a data set is another kind of average.

Exploration

Seven people were asked about their weekly income, and their responses form this data set:

$\$300,\$400,\$400,\$430,\$470,\$490,\$2900$$300,$400,$400,$430,$470,$490,$2900

The mean of this data set is $\frac{\$5390}{7}=\$770$$53907=$770, but this amount doesn't represent the data set very well. Six out of seven people earn much less than this.

Instead we can select the median, which is the middle score. We remove the biggest and the smallest scores:

$\$400,\$400,\$430,\$470,\$490$$400,$400,$430,$470,$490

Then the next biggest and the next smallest:

$\$400,\$430,\$470$$400,$430,$470

Then the next biggest and the next smallest:

$\$430$$430

There is only one number left, and this is the median - so for this data set the median is $\$430$$430. This weekly income is much closer to the other scores in the data set, and summarizes the set better.

It helped that this set was already in order, and that there were an odd number of scores. What happens when this isn't the case?

Exploration

Six people were asked to count the number of advertisements they saw while browsing the internet for an hour, and their responses form this data set:

$96,39,0,40,33,27$96,39,0,40,33,27

To find the median let's do the same thing we did before - we remove the biggest and the smallest scores:

$39,40,33,27$39,40,33,27

And the next biggest and the next smallest:

$39,33$39,33

Now that we are down to two scores, we find the number directly in between them. We can add the numbers together and divide by $2$2, just like finding a mean:

$\frac{39+33}{2}=\frac{72}{2}=36$39+332=722=36

The median number of advertisements that the six people saw was $36$36. This means that $50%$50% of people saw more than $36$36, and $50%$50% saw less than $36$36.

Finding the median of an ordered data set is much easier - if the set is scrambled up, you may want to rewrite it in order first.

Summary

The median of a numerical data set is the "middle" score, and its definition changes depending on the number of scores in the data set.

If there are an odd number of scores, the median will be the middle score.

If there are an even number of scores, the median will be the number in between the middle two scores, and half the scores will be greater than the median, and half will be less than the median.

 

 

Practice questions

Question 1

Find the mean of the following scores:

$6$6, $14$14, $10$10, $13$13, $5$5, $9$9, $14$14, $15$15

  1. Give your answer as a decimal.

Question 2

Find the median of the following scores:

$3,18,10,19,12,5,6,20,7$3,18,10,19,12,5,6,20,7

Outcomes

6.SP.A.2

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number

6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

6.SP.B.5.C

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered.

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