Statistical Displays

Step $1$	Determine the center (median) by finding the middle data point.
Step $2$	Find the maximum and minimum values on the graph. Use these values to calculate the spread (range) of the data.
Step $3$	Analyze the overall shape of the graph. Note any other interest features it may have.

Complete each step one at a time.

Step $1$

First, find the total number of data points in the set by counting all the dots in the given dot plot.

There are $15$ data points, which suggests that there were $15$ participants in the math competition that earned a score. The median of the data set is the $8\text{th}$ data point because it divides the set into two even halves. Locate this value on the dot plot.

The $8\text{th}$ data point is a $6,$ which means that the median, or the center, of the data set is $6.$ This measure indicates the middle value of all the scores earned by the participants.

Step $2$

The next step is to find the $\text{maximum}$ and $\text{minimum}$ values on the dot plot.

The minimum score earned by a participant is $2$ and the maximum score is $10.$ To find the range, calculate the difference between these values. The range of scores earned by the participants of the math competition is $8.$

Step $3$

The final step is to analyze the overall shape of the graph.

Notice that the left and right halves of the dot plot are not exactly identical. However, since they still resemble each other very closely, the plot can be considered symmetric in shape.

Step $1$	Title the plot based on the problem. Draw a number line to begin the dot plot, being sure to use values that are appropriate for the data set.
Step $2$	Determine the frequency of each value.
Step $3$	Place dots over each number on the number line that corresponds to the frequency for each value in the data set.

Complete each step one at a time.

Step $1$

First, think of a title for the dot plot. The given data set includes the scores of the participants from last year, so the title of the dot plot can be something like Scores From Last Year. Notice that the scores range from $1$ to $9.$ This means that the number line of the dot plot can be labeled with the consecutive integers from $1$ to $9.$

Step $2$

The next step is to determine the frequency of each score earned by the participants from last year's math competition. Count how many times each score from $1$ to $9$ appears in the set and write that number in a table next to the score.

Grade	Frequency
$1$	$1$
$2$	$0$
$3$	$1$
$4$	$2$
$5$	$0$
$6$	$3$
$7$	$3$
$8$	$2$
$9$	$1$

Step $3$

Lastly, place dots over each score from $1$ to $9$ equal to the number of times it appears in the data set. This is where the frequency table is helpful.

The dot plot for the given data set was successfully created.

Step $1$	Identify the independent and dependent variables.
Step $2$	List the frequency in each bar.
Step $3$	Interpret the data and describe the bar graph's shape. Use the interpretation to answer any questions about the data.

Complete each step one at a time.

Step $1$

First, the independent and dependent variables need to be identified. The horizontal line lists the days of a week, while the vertical line represents the average number of tickets sold.

The article analyzes how many tickets are sold on average on different days of a week and which day has the most sales. This means that the independent variable is the day of the week and the dependent variable is the average number of tickets sold on that day.

Step $2$

Next, the frequency in each bar should be listed and interpreted. Use the values given in the bar graph to indicate the height of each bar. Remember, the vertical line represents the average number of tickets sold on each day.

Day	Frequency
Monday	$64$
Tuesday	$70$
Wednesday	$62$
Thursday	$137$
Friday	$295$
Saturday	$342$
Sunday	$260$

Step $3$

Lastly, interpret the data.

Fridays and Saturdays have the highest numbers of average tickets sold, $295$ and $342$ respectively. The greatest number of tickets tends to be sold on Saturdays, when an average of $342$ tickets are sold.

Step $1$	Choose the number of intervals.
Step $2$	Determine the size of the intervals.
Step $3$	Make a frequency table.
Step $4$	Draw the histogram.

Complete each step one at a time.

Step $1$

The first step is determining how many intervals the histogram will have. All of the intervals must be the same length and the range covered by them must contain all of the data points in the set, so start by counting the number of values in the data set. To find a suitable number of intervals, take the square root of the number of data points. In this case, there are $30$ data points. This means that the histogram can have either five or six intervals. This solution will draw a histogram will have five intervals.

Step $2$

Next, the size of the intervals needs to be determined. This can be done by identifying the lowest and highest data values in the set. Since the lowest data value in the set is $7$ and the highest is $70,$ using five intervals with a range of $15$ will cover the numbers from $1$ to $75,$ which will include all data points. Then the intervals of the histogram will be the following.

Step $3$

The next step is to make a frequency table showing how many data points lie in each interval.

Interval	Data Points	Frequency
$1-15$	$7,$ $9,$ $11,$ $13,$ $15$	$5$
$16-30$	$17,$ $18,$ $19,$ $20,$ $21,$ $22,$ $23,$ $24,$ $24,$ $24,$ $25,$ $26,$ $27,$ $28,$ $30$	$15$
$31-45$	$32,$ $35,$ $37,$ $39,$ $42,$ $45$	$6$
$46-60$	$49,$ $55,$ $60$	$3$
$61-75$	$70$	$1$

Step $4$

The histogram can be constructed by drawing a bar over each interval with a height corresponding to the frequency listed in the table.

Recall what each part of a box plot represents.

Now consider the given box plot again.

By comparing the general box plot to this one, the minimum, maximum, median, first and third quartiles can be easily identified. The values alone are not very helpful, so to contextualize the values, consider what each of them means in the given context.

Concept	Value	Meaning
Minimum	$24$	The lowest level of pollution in the analyzed areas earned $24$ out of $100$ points.
Maximum	$98$	The highest level of pollution in the analyzed areas earned $98$ out of $100$ points.
Median	$63.5$	The average pollution score in the analyzed areas is $63.5$ out of $100.$
First Quartile	$39$	A quarter of the analyzed areas have a pollution score of $39$ or lower.
Third Quartile	$83$	A quarter of the analyzed areas have a pollution score of $83$ or higher.

Step $1$	Order the data set from least to greatest. Identify the minimum and maximum values.
Step $2$	Determine the median.
Step $3$	Determine the first and third quartiles.
Step $4$	Draw the box plot.

Complete each step one at a time.

Step $1$

Start by ordering the given data values from least to greatest. Now the minimum and maximum are easily identifiable in this ordered data set. Here, the minimum is $7$ and the maximum is $71.$

Step $2$

To find the median of the data set, count how many values there are in the data set. Now look for the value that lies in the middle of a data set. Since there are $20$ values, the median is the mean of the numbers in the $10\text{th}$ and $11\text{th}$ positions. The median is $37.$

Step $3$

The next step is to find the first and third quartiles of the data set. The median divides the set into two smaller sets, each with $10$ values. The first quartile is the middle of the first set. For this set, this means finding the mean of the $5\text{th}$ and $6\text{th}$ values. The third quartile is the median of the second set. For this half of the data set, the median is the mean of the $5\text{th}$ and $6\text{th}$ values.

Step $4$

To draw a box plot, organize the five-number summary of the data set. Mark the minimum and maximum values above a number line with two vertical segments, indicating the range of the box plot.

Next, mark the median with a vertical line segment inside the range above the number line. Remember that the line for the median falls inside the box.

The first and third quartiles are marked as the left and right sides of the box plot. The box plot can be completed by drawing a box between the quartiles and two horizontal segments between the left and right sides of the box and the minimum and maximum values.

The box plot is complete.

The data is fairly evenly distributed between the left and right side, so it is a roughly symmetric distribution. The value $5$ is a peak because it is the most frequently occurring value. The data values are clustered around $5.$

Next, decide which measures of center and spread are the most appropriate based on the symmetry of the distribution. Remember which measures can be used in each situation.

	Measure of Center	Measure of Spread
Symmetric Distribution	Mean	Mean absolute deviation
Non-Symmetric Distribution	Median	Interquartile range

Since this distribution is symmetric, it is best to use the mean as the measure of center and the mean absolute deviation as the measure of spread.

Here, the left side of the data is different than the right side, so the distribution is not symmetric. There are two clusters of data values within the intervals $17-20$ and $22-24,$ separated by a gap at $21.$ The peak of the data set is at $23.$

The most appropriate measures of center and spread can be determined by looking at the symmetry of the distribution. When the distribution is not symmetric, as in this case, it is best to use the median as the measure of center and the interquartile range as the measure of spread.

Hour	Distance Traveled (mi)
$1$	$70$
$2$	$135$
$3$	$203$
$4$	$278$
$5$	$348$

The horizontal axis can represent time in hours and the vertical axis can represent the distance traveled in miles. The distance data includes values from $70$ to $348,$ so a scale from $0$ to $420$ miles with an interval of $70$ miles is reasonable. Now the points can be plotted on a coordinate plane and connected with line segments.

The line graph representing the distance traveled by Tadeo's family is complete.

Notice that the graph shows an upward slant of the line with a steady increase from $1$ to $5$ hours. To predict about how long the drive to Tadeo's grandparents will take, follow the trend of the line to extend the graph to a distance of $420$ miles.

It can be predicted Tadeo's family will reach their destination after a total of about $6$ hours of driving.

Type of Display	Best Used to...
Bar Graph	$\dots$ show values corresponding to specific categories
Box Plot	$\dots$ show measures of spread for a data set
Dot Plot	$\dots$ show how many times each value occurs in the set
Histogram	$\dots$ show the frequency of data divided into equal intervals
Line Graph	$\dots$ show change over a period of time or in respect to a different quantity

The given data set lists countries and their corresponding life expectancies. After careful consideration of these types of displays, a bar graph looks like the best choice because it can show data that corresponds to specific categories — in this case, countries and life expectancies.

Country	Life Expectancy
United States	$76.3$
Japan	$84.5$
Germany	$80.9$
Brazil	$77.3$
China	$78.2$
India	$68.3$
Australia	$83.3$
South Africa	$62.4$

The countries can be marked on the horizontal axis and the life expectancy can be marked on the vertical axis. Next, draw bars with heights equal to the life expectancies.