4.02 Median and mode

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.

 

 

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).

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

3. 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

Question 2

 

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

Question 4

 

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$13−11). 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$90−2) which is quite a large range.

 

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

Practice questions

Question 6

Question 7