## Key Concepts

- Represent and interpret data in a dot plot
- Represent and interpret data in a histogram
- Represent and interpret data in a box plot
- Choose a data display

### Analyzing Data Displays

In your day-to-day life, you might have come across information, such as:

- Runs made by a batsman in the last 10 test matches.

- Number of wickets taken by a bowler in the last 10 ODIs.

- Marks scored by the students of your class in the mathematics unit test.

- Number of storybooks read by each of your friends, etc.

The information collected in all such cases is called data.

Data is usually collected in the context of a situation that we want to study.

For example, a teacher may like to know the average height of students in her class.

To find this, she will write the heights of all the students in her class, organize the data in a systematic manner and then interpret it accordingly.

Sometimes, data is represented graphically to give a clear idea of what it represents. Do you remember the different types of graphs which we have learned in earlier classes?

### Essential Question

What information about data sets can you get from different data displays?

**Data Analysis** is the process of bringing order and structure to collected data.

It turns data into information teams can use. Analysis is done using systematic methods to look for trends, groupings, or other relationships between different types of data.

**Data visualization** is the process of putting data into a chart, graph, or other visual formats that help inform analysis and interpretation. Data visuals present the analyzed data in ways that are accessible to and engage different stakeholders.

Data visuals are also used to communicate MEAL results to meet key stakeholder needs.

Multiple visuals will likely be needed to understand the larger change process and inform data use. Common data visual formats include:

- Dot plots
- Histograms
- Box Plots
- And many more

### Represent and Interpret Data in a Dot Plot

**Dot Plot: **

A dot plot is used to encode data in a dot or small circle.

The dot plot is shown on a number line that displays the distribution of numerical variables where a value is defined by each dot.

A frequency plot that shows the number of times a response occurred in a data set, where each data value is represented by a dot.

**Example 1 **

Manuel plans to buy a new car with the gas mileage shown.

To determine if the gas mileage of this car is good, he gathers data on the estimated city driving fuel efficiency, in miles per gallon, of several other cars.

How is the fuel efficiency of the car he wants to buy compared to the fuel efficiency of the other cars he researched?

**Example 1 **

**Solution:**

- Manuel wants to compare individual values, so a dot plot is a good way to show that information.
- Create a dot plot of the data by first drawing a number line that represents the range of the data.
- Plot each value from the table as a dot above the number line.

**Example 1 **

**Solution:**

- Use the dot plot to interpret the data.
- The dot plot shows that most of the values are clustered between 25 and 28.
- The car Manuel plans to buy has about the same city fuel efficiency as the other cars he has researched.

#### Try It!

**Example 1**

- What might account for the outlier?

**Solution:**

The outlier might be a high-end fuel-efficient car, so its performance is much greater than the other cars.

### Histogram

A histogram is a bar graph in which the data are organized into equal intervals. In the histogram, the horizontal axis shows the range of data values separated into measurement classes, and the vertical axis shows the number of values, or the frequency, in each class. Consider the histogram shown below.

**Example 2 **

A marketing team is about to launch a campaign for a new product that is targeted at adults aged 25-34 years. The team is researching the age range of viewers of a certain TV show to decide whether to advertise during the show.

The data show the ages of a random sample of 30 viewers of the show. Based on the findings, should the marketing team launch their campaign during this particular show?

**Example 2**

**Solution:**

Histograms are often used to represent data over ranges of numbers.

To create a histogram, first, decide on an appropriate interval for the data.

Create a frequency table to organize the data.

**Example 2:**

**Solution:**

Use the frequency table to create a histogram.

Use the histogram to interpret the data.

The histogram reveals that fewer than 25% of the viewers are between the ages of 25 and 35.

Based on this information, the marketing team should not launch their campaign during this particular show.

#### Try It!

**Example 2**

- What age group would be a good match for products advertised on this TV show? Explain.

**Solution:**

Step 1

The histogram shows that age groups 15-19 and 20-24 comprise about 67% of the viewers, so they should target viewers between the ages 15 and 24.

Step 2

Between the ages of 15 and 24 would be a good match for products advertised on TV show.

### Represent and Interpret Data in a Box Plot

A box plot—displays the five-number summary of a set of data.

The five-number summary is the minimum, first quartile, median, third quartile, and maximum.

In a box plot, we draw a box from the first quartile to the third quartile.

A vertical line goes through the box at the median.

The whiskers go from each quartile to the minimum or maximum.

**Example 3**

Students at a local high school organized a fundraiser for charity. Kaitlyn, the student council president, announces that more than half of the students raised over $50 each. The amounts of money raised by a random sample of 24 students are shown. Do the data support Kaitlyn’s claim?

**Example 3**

**Solution:**

Kaitlyn’s claim is about the distribution of values, so a box plot will reveal the information needed.

A box plot shows the distribution of data using a 5-number summary.

The **minimum** value is the least value in the set.

The **maximum** value is the greatest value in the set.

The **median** is the **middle **value in the set when the numbers are arranged from **least to greatest**. The **first quartile **is the **middle number **between the **minimum value and the median**.

The **third quartile **is the **middle number **between the **median and the maximum value**.

**Example 3**

**Solution:**

List the data from least to greatest to identify each of these values.

No. of observations n=24 Even

Step 1: n / 2 = 24 / 2=12^{th} Observation =45 And n/2+1=12+1=13^{th} Observation=45

Median =Average of n/2 and n/2+1 observations 45+45/2 = 45

Step 2: Similarly, the first quartile =32

Step 3: Similarly, the third quartile =52

**Example 3:**

**Solution:**

Create a box plot of the data.

Use the box plot to interpret the data.

The box plot shows that, based on the sample, the median amount of money collected was $45.

Since half of this population collected $45 or less, the data do not support Kaitlyn’s claim.

**Example 3**

#### Try It!

3. Suppose Kaitlyn wants to make the statement that 25% of the students raised over a certain amount. What is that amount? Explain.

**Solution:**

We need the top 25% of the data, which starts at the third quartile up to the maximum value.

Here, the third quartile is 52.

So, Kaitlyn can make a statement that 25% of the students raised over $52.

### Choose a Data Display

**Example 4 **

Helena’s dance team scores 68 points at a competition. The scores for all of the teams that competed are shown.

- Helena wants to know what place her team finished. Should Helena use a dot plot, histogram, or box plot to display the data?

- What place did Helena’s team finish in the competition?

**Solution**

- Compare the features of the three types of data displays.

- A
**dot plot**has dots for each value in the data. It shows clusters of data and outliers. - A
**histogram**groups values in a data set into ranges or intervals. Individual values are not displayed, but trends are observable. - A
**box plot**shows the center and spread of a data set.

A box plot does not show individual data but summarizes the data using 5 key pieces of information.

Helena is interested in displaying **individual scores**.

A **dot plot **will display the data in a way that is most helpful to Helena.

- Create and analyze a dot plot of the data.

A **dot plot **reveals that Helena’s team score was low compared to the other teams.

Only three teams scored lower, so Helena’s team is placed 13th out of 16.

#### Try it

Which data display should Helena use if she wants to know what percent of the teams scored higher than her team? Explain.

**Solution**

A histogram can be used since this groups the scores into intervals.

We then only need to find the total number of teams on the intervals who scored higher than Kaitlyn’s team and divide the total number of teams by 16.

### Histogram

- Make Sense and Persevere

Lucy usually pays between $0.40 and $0.60 per ounce for her favorite shampoo. She gathers prices of the same shampoo at different stores near her home. Prices are shown in dollars in the table. Create a data display that allows Lucy to easily compare the price she is paying to the other prices. How does the price she is currently paying compare?

**Solution:**

Step 1

Because there is an interval to what Lucy usually pays, a histogram is an appropriate display for the data set to create a histogram, first, decide on an appropriate interval for the data.

Remember that the intervals must be consistent.

**Create a frequency table to organize the data: **

Step 2

Use the frequency table to create a histogram:

Step 3

The histogram reveals that the prices of shampoos at different stores are between $0.80 to $1.00 per ounce, which are higher than what Lucy usually pays.

**Histogram**

The prices of shampoos at different stores are between $0.80 to $1.00 per ounce, which are higher than what Lucy usually pays.

### Check Your Knowledge

- What information about data sets can you get from different data displays?

- How is a dot plot different from a box plot? How are they similar?

- If you want to see data values grouped in intervals, which data display should you choose? Explain.

- Taylor says you can determine the mean of a data set from its box plot.

Is Taylor correct? Explain your reasoning.

- Can you determine the minimum and maximum values of a data set simply by looking at its dot plot? Histogram? Box plot? Explain.

- Use the data set shown for the exercises given above.

- Make a dot plot for the data. What information does the display reveal about the data set?

- Make a histogram for the data. What information does the display reveal about the data set?

- Make a box plot for the data. What information does the display reveal about the data set?

- What is the median value of the data set?

- What is the frequency of the data value 83?

**Answers**

**1. Dot plots display each data value from a set of data. **

They show clusters, gaps, and outliers in a data set.

Histograms do not show individual values but show clearly the shape of the data.

The data is organized into intervals. The bars show the frequency, or the number of times, that the data within that interval occur.

Box plots show the center (median) and spread of a distribution. Box plots provide the following information about a data set minimum, maximum, and median values, and the first and third quartile.

**2. A dot plot displays all of the data, while a box plot only shows the spread of a distribution of the data. **

They are similar in the sense that both display the minimum/least value and the maximum/greatest value in a data set

**3. A histogram should be used because using this data display; the data is organized into intervals. **

Histogram

**4. No. The median can be determined from the box plot, not the mean. **

The median is the middle value of the data set, while the mean is the

average of the data set, so they are different.

**5. Dot plot:**

Yes, each data is displayed in increasing order on a number line, so it is easy to see the minimum and maximum values.

**Histogram**:

No, the data is grouped into intervals, so we cannot really tell which among the first and last intervals are the minimum and maximum values, respectively.

**Box plot**:

Yes, a box plot gives a 5-number summary, two of which are the minimum and maximum values.

**6. i. **

Step 1

Create a dot plot of the data by first drawing a number line that represents the range of the data. Plot each value from the table as a dot above the number line.

Step 2

The dot plot shows that the data that occurred most is 8. The minimum value is I, and the maximum value is 15.

**6. ii. **

Step 2

Use the frequency table to create a histogram:

Step 3

The histogram reveals that most of the data are between 7 and 9, which is about 40% of the data

**6. iii. **

Step 1

A box plot shows the distribution of data using a 5-number summary.

The minimum value is the least value in the set.

The maximum value is the greatest value in the set.

The median is the middle value in the set when the numbers are arranged from least to greatest.

The first quartile is the middle number between the minimum value and the median.

The third quartile is the middle number between the median and the maximum value.

Step 2

List the data from least to greatest to identify each of these values. These are also the information revealed by the data set.

1, 4, 5, 7, 7, 8, 8, 8, 9, 10, 10, 11, 12, 13, 15

Minimum value: 1

First quartile: 1, 4, 5, 7, 7, 8, 8

Median: 8

Third quartile: 9, 10, 10, 11, 12, 13, 15

Maximum value: 15

Create a box plot of the data:

**7. Solution: **

List the data from least to greatest to identify each of these values.

These are also the information revealed by the data set:

32, 34, 35, 40, 40, 42, 42, 43, 43, 45, 46, 47, 47, 48, 49, 50

Minimum value: 32

First quartile: 40+40 / 2=40

Median: 43+43 / 2=43

Third quartile: 47+47 / 2=47

Maximum value: 50

Create a box plot of the data:

Step 3

The value of the median is 43.

**8. Solution: **

Step 1

A dot plot would be the most appropriate since it displays all the data.

Create a dot plot of the data by first drawing a number line that represents the range of the data.

Plot each value from the table as a dot above the number line.

Step 2

The data value 83 has a frequency of 5.

**Dot plot **shows the count of values within data sets.

**Histograms** show the distribution of values within a data set in ranges or intervals.

**Box plots **show the center and spread of a distribution using a five–number summary.

## Exercise

- Answer the following using the data given below

14,20,23,26,50,53,56,72,85,90,95,100,110,115

- What is the maximum value in the data?
- What are the first quartile and third quartile in the data?
- What is the median of the data?

- Plot the dot plot for the following data.

- Draw the histogram for the following data.

### Concept Map

### What have we learned

- Understand how to represent and interpret data in a dot plot
- Understand how to represent and interpret data in a histogram
- Understand how to represent and interpret data in a box plot
- Understand how to choose a data display

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