4.08 Review: Measures of center and spread

We have already learned about measures of center and spread .  Measures of central tendency attempt to summarize a set of data with a single value that describes the center or middle of the data values.  

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 value in a data set. 

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 and cross out the numbers on the ends until we get to the middle value.

Practice questions

Question 1

Question 2

 

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 piece of data. 

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 $1$1 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 values in a data set are very similar and clustered together, or whether there is a lot of variation in the data values 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 because we subtract the largest age from the smallest age ($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.

 

Practice questions

Question 5

Question 6