# 5.02 Review: Measures of center and spread

Lesson

Measures of central tendency attempt to summarize a set of data with a single value that describes the center or middle of the scores.

The three main measures of central tendency are the mean, median, and mode. Deciding which one is best depends on the characteristics of the particular set of data, as we already saw with the mean.

Measures of center

Mean

• The numerical average of a data set, this is the sum of the scores divided by the number of scores.
• Appropriate for sets of data where there are no values much higher or lower than those in the rest of the data set

Median

• The middle value of a data set ranked in order
•  A good choice when data sets have a couple of values much higher or lower than most of the others

Mode

• The data value that occurs most frequently
• A good descriptor to use when the set of data has some identical values, when data is non-numeric (categorical) or when data reflects the most popular item

## Median

The median is one way of describing the middle or the center of a data set using a single value. The median is the middle score in a data set.

### Which term is in the middle?

Suppose we have five numbers in our data set: $4$4, $11$11, $15$15, $20$20 and $24$24.

The median would be $15$15 because it is the value right in the middle. There are two numbers on either side of it.

$4,11,\editable{15},20,24$4,11,15,20,24

If we have an even number of terms, we will need to find the average of the middle two terms. Suppose we wanted to find the median of the set $2,3,6,9$2,3,6,9, we want the value halfway between $3$3 and $6$6. The average of $3$3 and $6$6 is $\frac{3+6}{2}=\frac{9}{2}$3+62=92, or $4.5$4.5, so the median is $4.5$4.5.

$2,3,\editable{4.5},6,9$2,3,4.5,6,9

If we have a larger data set, however, we may not be able to see right away which term is in the middle. We can use the "cross out" method.

### The "cross out" method

Once a data set is ordered, we can cross out numbers in pairs (one high number and one low number) until there is only one number left. Let's check out this process using an example. Here is a data set with nine numbers:

1. Check that the data is sorted in ascending order (i.e. in order from smallest to largest).

1. Cross out the smallest and the largest number, like so:

1. Repeat step 2, working from the outside in - taking the smallest number and the largest number each time until there is only one term left. We can see in this example that the median is $7$7:

Note that this process will only leave one term if there are an odd number of terms to start with. If there are an even number of terms, this process will leave two terms instead, if you cross them all out, you've gone too far! To find the median of a set with an even number of terms, we can then take the mean of these two remaining middle terms.

#### Practice questions

##### Question 1

Find the median of this set of scores:

$11$11, $11$11, $13$13, $14$14, $18$18, $22$22, $23$23, $25$25

##### Question 2

Given the following set of scores:

$65.2,64.3,71.6,63.2,45.2,62.2,46.8,58.7$65.2,64.3,71.6,63.2,45.2,62.2,46.8,58.7

1. Write the list of scores in ascending order.

2. Calculate the median.

## The mode

The mode is another measure of central tendency - that is, it's a third way of describing a value that represents the center of the data set. The mode describes the most frequently occurring score

Let's say we ask $10$10 people how many pets they have. $2$2 people say no pets, $6$6 people say one pet and $2$2 people say they have two pets. What is the most common number of pets for people to have? In this case, the most common number is one pet, because the largest number of people $\left(\frac{6}{10}\right)$(610) had one pet. So the mode of this data set is $1$1

#### Practice questions

##### Question 3

Find the mode of the following scores:

$2,2,6,7,7,7,7,11,11,11,13,13,16,16$2,2,6,7,7,7,7,11,11,11,13,13,16,16

1. Mode = $\editable{}$

##### Question 4

Find the mode of the following scores:

$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5

1. Mode = $\editable{}$

## Range

The range is a measure of spread in a numerical data set. In other words, it describes whether the scores in a data set are very similar and clustered together, or whether there is a lot of variation in the scores and they are very spread out.

If we looked at the range of ages of students in a $6$6th grade class, everyone would likely be between $11$11 and $13$13, so the range is $2$2 ($13-11$1311). This is quite a small range.

However, if we looked at the ages of people waiting at a bus stop, the youngest person might be a $2$2 year old and the oldest person might be a $90$90 year old. The range in this set of data is $88$88 ($90-2$902) which is quite a large range.

To calculate the range

Subtract the least score in the set from the greatest score in the set.

### Affecting the range

Remember, the range only changes if the greatest or least score is changed. Otherwise, it will remain the same.

#### Worked example

##### Question 5

Consider the set of data: $1,2,2,4,4,5,6,6,8,11$1,2,2,4,4,5,6,6,8,11. If the score of $8$8 is changed to a $9$9, how would the range be affected?

Think: What was the range when the score was an $8$8? What was the range when the score was changed to $11$11?

Do:

The range when the score was an $8$8 was $10$10 ($11-1$111). The range when the score was changed to $11$11 was also $10$10 ($11-1$111). Since the score that was changed was not the greatest or the least score, the range did not change.

#### Practice questions

##### Question 6

Find the range of the following set of scores:

$10,7,2,14,13,15,11,4$10,7,2,14,13,15,11,4

##### Question 7

The range of a set of scores is $8$8, and the greatest score is $19$19.

What is the least score in the set?