Summarize Data Distributions

Introduction: 

Describing a data set:  

To describe a data set, look at the shape and observe how the data are clustered or spread out. 

A distribution can be symmetric and clustered in the center: 

A distribution can show gaps, clusters, or outliers. It may spread out more to one side: 

8.7.1 Summarize a distribution that is symmetric 

Example 1: 

Marilyn tosses two six-sided number cubes 20 times. He records his results in a dot plot. 

How can you describe the data? 

Solution: 

Step1: 

To describe a data distribution, or how the data values are arranged, look at the overall shape. 

Step2: 

Since the data are roughly symmetric, the mean is the best measure of center. 

2+4+5+5+6+6+6+7+7+7+7+7+8+8+8+9+9+10+10+1220

  = 7.15 

The mean sum of tosses is 7.15 

Find the mean absolute deviation (MAD). 

5.15+3.15+2.15+2.15+1.15+1.15+1.15+0.15+0.15+0.15+0.15+0.15+0.85+0.85+0.85+1.85+1.85+2.85+2.85+4.85205.15+3.15+2.15+2.15+1.15+1.15+1.15+0.15+0.15+0.15+0.15+0.15+0.85+0.85+0.85+1.85+1.85+2.85+2.85+4.8520

= 33.620

= 1.68 

The mean absolute deviation is 1.68, so a typical sum of tosses is about 1.68 from the mean. 

Example 2: 

Frank measured the weights of seventeen different fruits and made the following dot plot. 

How Frank can describe the data? 

Solution: 

Step1: 

To describe a data distribution, or how the data values are arranged, look at the overall shape. 

Step2: 

Since the data are symmetric, the mean is the best measure of center. 

3+4+4+5+5+5+6+6+6+6+6+7+7+7+8+8+917 = 6 

The mean weight is 6. 

Find the mean absolute deviation (MAD). 

3+2+2+1+1+1+0+0+0+0+0+1+1+1+2+2+3/17 = 20/17

= 1.18 

The mean absolute deviation is 1.18, so a typical weight is about 1.18 from the mean. 

8.7.2 Summarize a distribution shown in a dot plot 

Example 3: 

Anna has conducted a survey on, “How long does it take to eat breakfast?” in grade 6. She noted the results in a dot plot. 

How can the data be used to describe the minutes took for eating the breakfast?  

Solution: 

Step1: 

Look at the distribution of the data in the dot plot. 

Step2: 

Because the data are not symmetric, the mean is not the best measure of center. Use the median and the IQR to describe the data distribution.  

Median = 3 

IQR = 5 – 1 = 4 

The time taken by at least half of the students is between 5 minutes and 1 minute. The typical student has taken 4 minutes of time to take breakfast. 

8.7.3 Summarize a distribution shown in a box plot 

Example 4: 

Ms. Laura collected information about the students with dogs in each classroom at her school. Her box plot looks like below: 

How can you summarize these data? 

Solution: 

First, look at the overall shape. Then find the measures of center and variability. 

• The data are spread out equally. 
• The data is symmetric. 
• The mean is the center of the data. 
• The first quartile is 9 students and third quartile is 21 students. 

• The interquartile range is 12 students. 

Example 5: 

The box plot shows the recorded speeds of cars traveling on a city road. How can you summarize these data? 

 

Solution: 

First, look at the overall shape. Then find the measures of center and variability. 

• The data are spread out to the right. 
• The median is the center of the data. 
• The recorded speeds range from 19 miles per hour to 40 miles per hour.  
• The middle half of the data range from 22 miles per hour to 34 miles per hour. Because the boxes are longer than the whiskers, there is more variation among the middle half of the data. Having more variation means there is a lesser consistency among the middle 50% of the data than in either whisker. 

• The first quartile is 22 mph and third quartile is 34 mph. 
• The interquartile range is 12 mph. 

Exercise:

1.Fill in the blanks.

a. Use the mean and the mean absolute deviation (MAD) to describe the data when the data are _______________________.

b. Use the median and the IQR to describe the data when the data are _________________.

2. The following dot plot shows the songs on each album in Nancy’s collection. 
How can the data be described.

3. The following dot plot shows the numbers of brothers and sisters for pupil in a class. How can you describe the data?

4. The following dot plot shows the Spanish quiz scores of a class.

Describe the shape of the data.

5. The following dot plot shows the number of miles students of class 6 ran in a week.

Describe the shape of the data.

6. The following dot plot shows the number of miles students of class 9 ran in a week.

Describe the shape of the data.

7. The average gas mileage, in miles per gallon, for various sedans is surveyed and shown in the box plot. Describe the distribution of the data.

8. Earthquakes occur at different depths below Earth’s surface. Stronger earthquakes happen at depths that are closer to the surface. The table shows the depths of recent earthquakes, in kilometers.

Describe the distribution of the data.

9. The box plot shows the ages of vice presidents when they took office. Describe the distribution of the data. What does it tell you about the ages of vice presidents.

10. The ages of children taking a hip-hop dance class are 10, 9, 9, 7, 12, 14, 14, 9, and 16 years old. Construct a box plot of the data. Then describe the distribution of the data.

Concept Map: 

What have we learned:

• Summarize a data distribution that is symmetric.
• Summarize a data distribution that is non symmetric.
• Use different measures of center of data and variability depending on the distribution of the data.