Display, Describe and Summarize Data

Key Concepts

• Choose the best measure of center to describe a data set.
• Choose the best measure of variability to describe a data set.

Choose the appropriate statistical measures

Outlier

An outlier is a data value that is very far away from the other data values. It can be much greater in value or much less than the other values.

Consider the data set shown.

225, 245, 295, 305, 360, 387, 388, 420, 470, 480, 625, 780

How do you know if either of the extreme values, 225 or 780, are considered outliers?

Most of the values are around 400. The value of 780 is an outlier because it is much bigger than most of the other values.

Example 1:

The ages, in years, of the candidates in an election are 55, 49, 48, 57, 23, 63, and 72. Identify any outliers in the data set.

Solution:

The value of 23 is much lower than the rest of the data. The outlier of the data is 23.

Example 2:

In the below graph, the outlier lies on the far left.

The value in the month of January is significantly less than the values in the other months of the graph.

Some other examples of outliers:

Choose the best measure of center to describe a data set

Example 3:

Marilyn tracked the daily high temperatures for nine days and the results are recorded in the table shown below.

What is the good choice to describe the center of the data set?

Solution:

If you observe the above data set, 76 is an outlier. It lies outside most of the other values in the data set.

Let us calculate the mean.

Mean =  44+43+42+40+42+45+39+38+18944+43+42+40+42+45+39+38+189 / 9

= 351/9

          = 67.55

The outlier causes the mean to be greater than the values of the largest group of data.

Example 4:

The following are the scores of a basketball team for one season.

44, 43, 42, 40, 42, 45, 39, 38, 18.

Is the mean or median the best measure of center of the data? Explain.

Solution:

If you observe the above data set, 18 is an outlier. It lies outside most of the other values in the data set.

Let us calculate the mean.

Mean = 39

The outlier causes the mean to be less than the values of the largest group of data.

Choose the best measure of variability to describe a data set

Example 5:

The following are the scores of a basketball team for one season.

44, 43, 42, 40, 42, 45, 39, 38, 18.

Find the measure of variability that best describes the data set.

Solution:

If you observe the above data set, 18 is an outlier. It lies outside most of the other values in the data set. So, the median is a better measure of center for scores than the mean.

Example 6:

The following data set shows the amount of money raised by the homerooms for grade levels at a middle school.

88, 116, 94, 108, 112, 124

Find the measure of variability that best describes the data set.

Solution:

The range is from 88 to 124, so there is no outlier. The mean, 107, is a good measure of center for the data.

MAD =  19+9+13+1+5+17 / 6

= 64/6

= 10.66

Exercise

1. Fill in the blanks:
   • The ________________ is a good choice to describe the center of a data set when the data are clustered together.
   • The ________________ is a good choice to describe the center of a data set when the data contains an outlier.
2. Fill in the blanks:
   • The _________________ is a good choice to describe the variability when you are using the mean.
   • The _________________ is a good choice to describe the variability when you are using the median.
3. The lengths, in feet, of various bridges are 354, 88, 251, 275, 727. Identify any outliers in the data.
4. Marilyn has conducted a survey on the lifespan of some animals. She made the following tables with her findings.

Average Lifespan

Animal	Lifespan (years)
Elephant	35
Dolphin	30
Chimpanzee	50
Tortoise	200
Gorilla	30
Gray Whale	70
Horse	20

Choose the measure that best describes the center.

• Find the best measure of variability for the above data.
• The frequency table shows the number of states that border each state in the United States.

Bordering States	Frequency
0 – 1	3
2 – 3	13
4 – 5	22
6 – 7	10
8 – 9	2

What are the most appropriate measures to describe the center and the variation?

• Find the most appropriate measures to describe the center and the variation of the following data set.

42, 68, 71, 72, 72, 74, 75

• Find the most appropriate measures to describe the center and the variation of the following data set.

35, 30, 50, 200, 30, 70, 20

• The ages, in years, of employees in an office are 55, 49, 48, 57, 23, 63, and 72. Identify any outliers in the data set. Justify your response.
• The table shows the cooking temperatures for different recipes.

175, 325, 325, 350, 350, 350, 400, 450

Identify any outliers in the data set. Justify your response.