Mathematics 1040 Statistics
Chapter 3 Activity 1 (Sec. 3.2)
MEASURES OF CENTER
Names: ________________________
_________________________
_________________________
_________________________
Objectives: Be able to compute and understand the meaning of each of the mean, median, mode, and
midrange for a data set. Be able to compute the mean of a grouped data set.
When we have a (large) data set, we often need to describe or “summarize” it. One way important
characteristic of a data set is its center or middle. However there are several different ways to compute
/ talk about this center or middle, each with its own strengths and weaknesses. This activity will
discuss the most common measures of center and ask you to apply what you learned.
Let’s first review two important terms:
Population:
Sample:
Mean
The most commonly used measure of “center” is the mean (technically the “arithmetic mean” but
often referred to as the “average”). How do you find the mean? Write out the steps in the space below:
The symbol for population mean is  (“mu”) and for sample mean it is 𝑥̅ (“x-bar”). Using the correct
symbol, find the mean for the following data set, (a random sample of reported amount students paid
for their stats book this semester): {35, 40, 60, 15, 15}. Your answer should be “ = #” OR “𝑥̅ = #.”
The next randomly selected student paid $50 for the text. Find the mean for the new data set
{35, 40, 60, 15, 15, 50}.
How did the mean change? Why did it change that way?
The next randomly selected student paid $130 for his/her book. What is the mean of this new dataset:
{35, 40, 60, 15, 15, 50, 130}
How did the mean change? Why did it change that way?
What does the mean signify? In other words, what does the mean tell us?
In general, what are some advantages of the mean?
Disadvantage(s) of the mean?
Created by Jonathan Bodrero.
For non-commercial educational use.
Median
Another commonly used measure of “center” is the median. How do you find the median for a data
set? Write out the steps:
Find the median for each of the data sets.
{35, 40, 60, 15, 15}
{35, 40, 60, 15, 15, 50}
{35, 40, 60, 15, 15, 50, 130}
Compared to the mean, did the median change more or less as we added additional values?
What does the median signify? In other words, what does the median tell us?
In general, what are some advantages of the median?
Disadvantage(s) of the median?
Mode
The next measure of “center” we’ll study is the mode. How do you find the mode for a data set?
Write out the steps:
Find the mode for each of the data sets.
{35, 40, 60, 15, 15}
{35, 40, 60, 15, 15, 50}
{35, 40, 60, 15, 15, 50, 130}
Find the mode for the following data sets:
{35, 40, 60, 15}
{35, 40, 60, 15, 15, 60}
What does the mode signify? In other words, what does the mode tell us?
In general, what are some advantages of the mode?
Disadvantage(s) of the mode?
Created by Jonathan Bodrero.
For non-commercial educational use.
Midrange
The last measure of “center” that we will discuss (briefly) is the midrange (although it is rarely used).
How do you find the midrange for a data set? Write out the steps:
Find the midrange for each of the data sets.
{35, 40, 60, 15, 15}
{35, 40, 60, 15, 15, 50}
{35, 40, 60, 15, 15, 50, 130}
What does the midrange signify? In other words, what does the midrange tell us?
In general, what is the advantage of the midrange?
Disadvantages of the midrange?
In real life, we generally have data sets that are larger than 5 or 7 points. The calculator can help us
find some of these measures of middle. Below is a random sample of 25 cost of stats textbooks
(rounded to the nearest dollar) reported from Prof. Bodrero’s students this semester.
50, 6, 20, 82, 30, 30, 5, 30, 0, 8, 20, 12, 17, 50, 40, 40, 0, 5, 6, 15, 82, 130, 20, 20, 37
Enter this data into your calculator by pushing the STAT button and choosing the EDIT option. Go to
List 1 (L1) and enter each of the values. [If there is already data in L1, clear it out by using arrows to
go up to the L1 icon and pushing the “CLEAR” button, not the “DEL” button.] Then push the colored
“2nd” button and the MODE button next to it to QUIT out to the home screen.
Next, push the STAT button and move over to the CALC option. Run option 1: 1-Var Stats and tell it
to use List 1 (2nd button then 1 key is a shortcut for “L1”). Write what you see below. Then find each
of the four measures of middle (mean, median, mode, midrange) for this sample of data.
Mean of Grouped Data
Sometimes we have data reported in a frequency distribution and we need to find the mean. Even if
the data is grouped into classes, (so we don’t know each original data value) we can still approximate
the mean to a fairly high degree of accuracy. Let’s look at an important topic: US unemployment.
Unemployment rates for the 50 states and DC. (Source US Bureau of Labor Statistics, 2015)
Rate (%)
2.0 – 2.9
3.0 – 3.9
4.0 – 4.9
5.0 – 5.9
6.0 – 6.9
7.0 – 7.9
Frequency
1
7
14
17
11
1
Class midpt.
(Note: North Dakota = 2.7%, Utah = 3.6%, California = 6.2%, Nevada = 6.8%, W. Virginia = 7.5%)
Created by Jonathan Bodrero.
For non-commercial educational use.
To compute the mean on data that has already been grouped, we assume that each value is at the
midpoint of its class. (This is a pretty safe assumption as the higher values and the lower values within
each category tend to average out to the class midpoint). Fill in the class midpoints in the table above.
Next we do the average on all 51 data points. We could do 2.45 + 3.45 + 3.45 + 3.45 + 3.45 +… and
then divide by 51. But since many of them are repeated, we can save a lot of time another way. Take
the first class midpoint, 2.45, and multiply by its frequency, 1, to get 2.45. Do the same for each of the
classes and add them all up. Then divide by 51 (not 6) to get the mean. This is shown by the formula
𝑥̅ =
∑(𝑓∙𝑥)
∑𝑓
where x represents each class midpoint and f is the corresponding frequency. Use this
method to compute the mean unemployment rate for states (round to 2 decimal places).
How does your computed value compare with the mean (using every actual data value) of 5.13?
The unemployment rate in the US (as a whole) is 5.3%. Why are the numbers computed above off?
The mean of a frequency distribution can be calculated with your TI-83/84. First, go to the STATS
menu and EDIT. Put the class midpoints in the first list, L1 and the corresponding frequencies in L2.
Then QUIT and do STAT then CALC and 1:1-Var Stats L1, L2. Write what you see below.
Skewness
A distribution of data is skewed if it is not symmetric and extends more to one side than the other.
Left skewed (negatively skewed)
Symmetric
Right skewed (positively skewed)
Which shape does the unemployment data most closely resemble, left skewed, symmetric, or right
skewed?
Created by Jonathan Bodrero.
For non-commercial educational use.