## Key Concepts

- Understand how to interpret the shapes of data displays
- Understand what is symmetric data display
- Understand what is skewed data display
- Understand how to determine a display whether it is symmetric or skewed.

### Interpret the Shape of a Distribution

How does the shape of a data set help you understand the data?

**Example 1 **

The histograms show the weights of all the dogs entered in two different categories in a dog show. Consider each data set. What inferences can you make based on the shape of the data?

The histogram is symmetric and shows the data are evenly distributed around the centre.

The mean and median of the data are equal or almost equal.

Based on the data, you can infer that most of the dogs in this category weigh between 30 and 59 pounds.

The histogram shows the data are skewed right. The mean is greater than the median.

Based on the data, you can infer that most of the dogs in this category weigh less than 30 pounds.

#### Try It!

1. **Suppose a third category of dogs has a mean of 40 lb and a median of 32 lb. **

What can you infer about the shape of the histogram for dogs in this category?

Solution:

- Since the median is smaller than the mean weight of third category, the data would be right-skewed.

### Interpret the Shape of a Skewed Data Display

**Example 2 **

Display customers of a bagel shop complained that some bagels weigh less than the amount on the label. A quality control manager randomly sampled 30 bagels and weighed them. Based on this sample, is a change in the process for making bagels warranted?

The shape of the data display is skewed left. That is common when the mean of a data set is less than the median of a data set.

Because the mean of the sample is less than the advertised weight, the quality control manager wants to recommend some changes to the production process. However, because the sample consists of only 30 bagels, the manager decides to generate another random sample.

#### Try It!

2. **How do skewed data affect the mean in this context?**

Solution:

The skewed data affects the mean in two ways. They are:

- If the shape of the data display is skewed left, the mean of the data set is less than the median

of the set.

- If the shape of the data display is skewed right, the mean of the data set is more than the median of the set.

### Compare Shapes of Skewed Data Displays

**Example 3 **

The manager generates a second random sample of 30 bagels. Is a change in the process for making bagels warranted based on this sample? How does this sample compare to the previous sample?

The shape of this data display is skewed right.

This often occurs when the mean of the data is greater than the median of the data.

In this example, the mean is greater than the advertised weight and greater than the mean weight from the first sample.

Based only on this sample, the manager could recommend changes.

However, because this sample gives opposite results to the first sample, the manager now has conflicting findings of the mean weights of the bagels.

#### Try It!

3. **What does the shape of the histogram for the second sample tell you about the data?**

Solution:

The shape of the data second sample is right-skewed because there is a high frequency on the left side and tail on the right side. As a result, we would find that the mean is to the right of the median of sample data.

Histogram tells that the sample distribution is right-skewed and the mean is to the right of the median of sample data.

### Interpret the Shape of a Symmetric Data Display

**Example 4 **

The quality control manager generates a third random sample that contains twice as many data points. A histogram that represents this third, larger sample is shown below. Based on this sample, is a change in the process for making bagels warranted?

The data points are symmetrically distributed around the centre.

In this sample, the mean and median weights are the same.

Based on this larger sample, the quality control manager determines that the process of making bagels does not need to change because the data are centered around the advertised weight of 90 grams.

#### Try It!

4. **Suppose the quality control manager adds another 10 bagels to the third sample. If 5 of the bagels are 78 g each, and 5 of the bagels are 106 g each, would that affect the mean and median weights?**** Explain.**

Solution:

Adding 5 values at each end of the distribution does not affect the median. Thus, the median is still 90 grams.

The mean is 90 grams for the third sample. Since, the 106 grams is farther from the mean than 78 grams, adding 5 units of 78 and 106 grams, would shift the mean towards the right and increase the mean

Value.

Mean increases but median remains unchanged.

### Comparing the Shapes of Data Sets

**Example 5 **

Jennifer is considering job offers from three different school districts. The histograms show the salary ranges for similar positions in each school district. What do the shapes of the data tell Jennifer about the teacher salaries in each district?

The data from teachers with similar positions in school district 101 have salaries that centre around the mean. This indicates that the salaries of the teachers are fairly evenly distributed.

The data from teachers in school district 201 is skewed right. Most of the teachers in that district have salaries that are lower than the mean. This indicates that there are only a few teachers that make a higher salary.

The data from teachers in school district 301 is skewed left. Most of the teachers with similar positions have salaries that are higher than the mean. This indicates that there are more teachers that make a higher salary.

If Jennifer’s starting salary is the same at each school district, she should strongly consider school district 101 or school district 301. Based on the data represented here, these districts have more potential for Jennifer to advance her salary.

#### Try It!

5. **Suppose a fourth school district offers Jennifer a job. School district 401 has a mean salary of $57,000 and a median salary of $49,000. Should Jennifer consider accepting the job offer with school district 401? Explain.**

Solution:

Most of the teachers in the district 401 have salaries that are lower than the mean. This indicates that there are only a few teachers that make a higher salary.

- So, Jennifer rejected the job offer because a few teachers only are making a high salary.

### Concept Summary

Interpreting the Shapes of Data Displays

- The shape of a data display reveals a lot about the data set. In a symmetric data display, the data points are evenly spread on either side of the center. The mean is equal (or approximately equal) to the median. In a skewed data display, the data points are unevenly spread on either side of the center (median). A data display can be skewed right or skewed left. The mean and median are not equal.

#### Graphs

- This data set is symmetric. There are a similar number of data points greater than and less than the mean.

- One family with 5 pets skewed the data to the right. The very small sample size makes inferences unreliable.

### Check your knowledge

- The displays represent house prices in a town over two consecutive years. What do the displays tell you about the change in house prices in the two years?

- Tell whether each display is skewed left, skewed right, or symmetric. Interpret what the display tells you about the data set.

### Answers

1.

Solution:

- In the first year, the house prices were left-skewed as the histogram has a tail on the left side. This means that more houses had higher prices.
- In the next period, the display for house prices became skewed. The mean would be close to median. We observe that the number of houses priced lower increases, while those with higher prices decrease.
- As a result of this, mean house price decreases from first to second year.

2. a

Solution:

Since there is a low frequency on the left side, the data is left-skewed.

As it is left-skewed, the mean would be less than the median.

Left skewed, mean would be less than the median.

b.

In the display, we can see that the box plot is symmetrical around the median.

It suggests the data is symmetric. The data points are evenly spread on either side of the center.

- Symmetric, data points are spread evenly on either side of the median.

c.

Solution:

The histogram is asymmetric with a higher frequency on the left and lower frequency on right side.

Thus, the display is right-skewed.

As a result, the mean would be greater than the median.

- Right-skewed, the mean greater than the median.

## Exercise

- The frequency table shows the number of raffle tickets sold by students in your grade. Display the data in a histogram. Describe the shape of the distribution.

- A police officer measures the speeds (in miles per hour) of 30 motorists. The results are shown in the table.
- Display the data in a histogram using six intervals beginning with 31-35.
- Which measures of center and variation best represent the data?

### Concept map

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