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Histograms

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Histograms

Remember how Jasper made a frequency table in the Frequency Tables to Organize and Display Data Concept? Well, now he is going to take this frequency table and try to make a histogram. Take a look.

Jasper is curious about how many days it takes a musher to finish the Iditarod. Looking online, he has discovered that the average is from 10 – 15 days, but that isn’t specific enough for him.

“I want to know more details about it,” he tells Mr. Hawkins first thing on Monday morning.

“Well, you have to narrow down your findings. I would suggest you look at the final standings from 2010. Then you can create a frequency table and a histogram.”

“Alright, that’s a good idea,” Jasper says.

Jasper begins his research on the Iditarod website. He makes notes on the number of days that it took the mushers in the 2010 Iditarod to finish. Here is the frequency table that he created with his findings.

Days Tally Frequency
8 I I
9 I I I I I 18
I I I I I
I I I I I
I I I
10 I I I I I 16
I I I I I
I I I I I
I
I I I I I I I 6
I
12 I I I I I 9
I I I I
13 I I I I 4

Next, Jasper began making his histogram. But as soon as he started to draw it, something did not look right.

Jasper could use some help. In this Concept, you will learn how to take a frequency table and make a histogram out of it. Pay close attention and at the end of this Concept you will be able to help Jasper create his visual display.

Guidance

Frequency tables are a great way to record and organize data. Once you have created a frequency table, we can make a histogram to present a visual display of the information in the frequency table.

What is a histogram?

A histogram shows the frequency of data values on a graph. Like a frequency table, data is grouped in intervals of equal size that do not overlap. Like a bar graph, the height of each bar depicts the frequency of the data values. A histogram differs from a bar graph in that the vertical columns are drawn with no space in between them.

Now let’s look at creating a histogram from a frequency table.

Create a histogram using the results on the frequency table below.

Hours Slept Each Night
Number of Hours Slept Tally Frequency
5 I 1
6 I I 2
7 I I I I 4
8 I I I 3
9 I I I 3
10 I I I 3
11 I I 2
12 I I 2

To create a histogram:

1. Draw the horizontal (x) and vertical (y) axis.

2. Give the graph the title “Hours Slept Each Night.”

3. Label the horizontal axis “Hours.” List the intervals across the horizontal axis.

4. Label the vertical axis “Frequency.” Since the range in frequencies is not that great, label the axis by ones.

5. For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. On a histogram, there is no space in between vertical columns.

Take a few minutes to copy down the steps for creating a histogram in your notebook.

Create a histogram to display the data on the frequency table below.

Number of Minutes on the Computer Tally Frequency
0 – 5 I I I 3
6 – 10 I I 2
11 – 15 I I I 3
16 – 20 I I 2
21 – 25 I 1
26 – 30 I 1
31 – 35 I 1
36 – 40 I 1
41 – 45 I I 2
46 – 50 I 1
51 – 55 I 1
56 – 60 I I 2

To create a histogram:

1. Draw the horizontal (x) and vertical (y) axis.

2. Give the graph the title “Minutes Spent on the Computer.”

3. Label the horizontal axis “Minutes.” List the intervals across the horizontal axis.

4. Title the vertical axis “Frequency.” Label the axis by halves (0.5).

5. For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. Recall that on a histogram, there are no spaces in between vertical columns.

Sometimes, you will be given a set of data that you will need to organize. This data will be unorganized. To work with it, you will have to organize it by creating a frequency table. Then you can use that frequency table to create a histogram.

Fifteen people were asked to state the number of hours they exercise in a seven day period. The results of the survey are listed below. Make a frequency table and histogram to display the data.

8, 2, 4, 7.5, 10, 11, 5, 6, 8, 12, 11, 9, 6.5, 10.5, 13

First arrange the data on a frequency table. Recall that a table with three columns needs to be drawn: one for intervals, one for tallied results, and another for frequency results. The range in values for this set of data is eleven. Therefore, data will be tallied in intervals of three.

Hours of Exercise Tally Frequency
0 – 2 I 1
3 – 5 I I 2
6 – 8 I I I I I 5
9 – 11 I I I I I 5
12 – 14 I I 2

Next, the data needs to be displayed on a histogram. Recall that a horizontal (x) and vertical (y) axis needs to be drawn. List the intervals across the horizontal axis. Name this axis “Hours of Exercise.” Label the vertical axis by ones. Title the vertical axis “Frequency.” For each set of intervals, draw vertical columns the appropriate frequency. Color in the vertical columns and ensure that no space is between them. Title the graph “Hours of Exercise.”

Now let’s make some conclusions based on the information displayed in the histogram.

Looking at the histogram above, you can that equal numbers of people reported that they exercise between six and eight and nine and eleven hours each week. Two people stated that they exercise between three and five hours per week. Two people reported that they exercise between twelve and fourteen hours per week. Zero to two is the hours with the least frequency.

Look at this frequency table and use it to complete the following questions.

Number of Sodas Tally Frequency
0 – 3 I I I I I I I I 8
4 – 7 I I I I I I I 7
8 – 11 I I I 3
12 – 15 I I 2

Example A

Which category is the most popular?

Solution: 0 - 3 Sodas

Example B

Which category is the least popular?

Solution: 12 - 15 sodas

Example C

What is the difference between the greatest number of sodas and the least?

Solution: 8 - 3 = 5

Now back to Jasper and the histogram.

Here is the original problem once again. Reread it and then look at the histogram created from the frequency table.

Jasper is curious about how many days it takes a musher to finish the Iditarod. Looking online, he has discovered that the average is from 10 – 15 days, but that isn’t specific enough for him.

“I want to know more details about it,” he tells Mr. Hawkins first thing on Monday morning.

“Well, you have to narrow down your findings. I would suggest you look at the final standings from 2010. Then you can create a frequency table and a histogram.”

“Alright, that’s a good idea,” Jasper says.

Jasper begins his research on the Iditarod website. He makes notes on the number of days that it took the mushers in the 2010 Iditarod to finish. Here is the frequency table that he created with his findings.

Days Tally Frequency
8 1 1
9 11111 18
11111
11111
111
10 11111 16
11111
11111
1
11 11111 6
1
12 11111 9
1111
13 1111 4

Next, Jasper began making his histogram. But as soon as he started to draw it, something did not look right.

Then Jasper began to notice that he needed to put the number of mushers on the y axis and the number of days on the x axis. He included each day instead of a range of days since there were only six possible options for days to finish.

Here is Jasper’s final histogram.

Guided Practice

Here is one for you to try on your own.

The data on the table below depicts the height (in meters) a ball bounces after being dropped from different heights. Create a frequency table and histogram to display the data.

6 \quad 9 \quad 4 \quad 12 \quad 11 \quad 5 \quad 7 \quad 9 \quad 13 \quad 5 \quad 6 \quad 10 \quad 14 \quad 7 \quad 8

Answer

First arrange the data on a frequency table.

Recall that a table with three columns needs to be drawn: one for intervals, one for tallied results, and another for frequency results. The range in values for this set of data is nine. Therefore, data will be tallied in intervals of two.

Bounce Height Tally Frequency
3 – 4 I 1
5 – 6 I I I I 4
7 – 8 I I I 3
9 – 10 I I I 3
11 – 12 I I 2
13 – 14 I I 2

Next, the data needs to be displayed on a histogram.

Recall that a horizontal (x) and vertical (y) axis needs to be drawn. List the intervals across the horizontal axis. Name this axis “Bounce Height.” Label the vertical axis by ones. Title the vertical axis “Frequency.” For each set of intervals, draw vertical columns the appropriate frequency. Color in the vertical columns and ensure that no space is between them. Title the graph “Bounce Heights.”

Now what conclusions can we draw from the frequency table and histogram?

You can see that the most frequent bounce heights were between five and six meters. The least frequent bounce heights were between three and four meters. Three balls bounced between seven and eight meters and nine and ten meters. Two balls bounced between eleven and twelve meters and thirteen and fourteen meters.

Video Review

This is a video on frequency tables and histograms.

Explore More

Directions : Use what you have learned to complete each dilemma.

1. Create a histogram to display the data from the frequency table below.

Monthly Internet Purchases
Data Values Tally Frequency
0 – 3 I I I 3
4 – 7 I I I I 4
8 – 11 I 1
12 – 15 I I 2

2. The data collected depicts the number of letters in the last names of twenty people. Create a frequency table to display the data.

12 \quad 3 \quad 5 \quad 9 \quad 11 \quad 2 \quad 7 \quad 5 \quad 6 \quad 8 \quad	14 \quad 4 \quad 8 \quad 7 \quad 5 \quad 10 \quad 5 \quad 9 \quad 7 \quad 15

3. Create a histogram to display the data.

4. The data collected depicts the number of hours twelve families traveled this summer to their vacation destination. Create a frequency table to display the data.

7 \quad 3 \quad 10 \quad 5 \quad 12 \quad 9 \quad 8 \quad 4 \quad 3 \quad 11 \quad 3 \quad 9

5. Create a histogram to display the data.

6. Write a few sentences to explain any conclusions that you can draw from the data.

7. Generate a question that you will use to survey twenty people.

8. Make a table to collect the answers.

9. Display the data on a frequency table

10. Create a histogram to display the data histogram.

Here is a list of the number of students who did not complete their homework in one month.

1, 1, 3, 3, 4, 3, 3, 5, 6, 1, 1, 1, 2, 2, 3

11. Create a frequency table of the data.

12. What is the most popular value?

13. What is the least popular value?

14. What is the range of values?

15. What is the average?

Vocabulary

bar chart

bar chart

A bar chart is a graphic display of categorical variables that uses bars to represent the frequency of the count in each category.
bar graph

bar graph

A bar graph is a plot made of bars whose heights (vertical bars) or lengths (horizontal bars) represent the frequencies of each category, with space between each bar.
bell curve

bell curve

A normal distribution curve is also known as a bell curve.
bell shaped

bell shaped

A bell shaped histogram is a histogram with a prominent ‘mound’ in the center and similar tapering to the left and right.
binning

binning

Binning involves separating your data separated into separate classes or categories.
bins

bins

Bins are groups of data plotted on the x-axis.
class limits

class limits

Class limits are, collectively, the upper and lower limit of an interval.
class mark

class mark

A class mark is the middle value, or average of the class limits.
extreme outliers

extreme outliers

Extreme outliers include points more than 3 times the middle half of your data.      .
frequency density

frequency density

The vertical axis of a histogram is labelled frequency density.
frequency distribution table

frequency distribution table

A frequency distribution table lists the data values, as well as the number of times each value appears in the data set.
frequency polygon

frequency polygon

A frequency polygon is a graph constructed by using lines to join the midpoints of each interval, or bin.
Frequency table

Frequency table

A frequency table is a table that summarizes a data set by stating the number of times each value occurs within the data set.
Histogram

Histogram

A histogram is a display that indicates the frequency of specified ranges of continuous data values on a graph in the form of immediately adjacent bars.
Interval

Interval

An interval is a range of data in a data set.
left-skewed distribution

left-skewed distribution

A left-skewed distribution has a peak to the right of the distribution and data values that taper off to the left.
mild outliers

mild outliers

Mild outliers include data points that are more than 1.5 times the middle half of your data above the upper, or below the lower, quartiles.
multimodal

multimodal

When a set of data has more than 2 values that occur with the same greatest frequency, the set is called multimodal    .
normal distributed

normal distributed

If data is normally distributed, the data set creates a symmetric histogram that looks like a bell.
Outlier

Outlier

In statistics, an outlier is a data value that is far from other data values.
Range

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.
relative cumulative frequency plot (ogive plot)

relative cumulative frequency plot (ogive plot)

A relative cumulative frequency plot, or  ogive plot, shows how the data accumulate across the different values of the variable.
relative frequency histogram

relative frequency histogram

A relative cumulative frequency histogram is a histogram except the vertical bars as the relative cumulative frequencies.
right-skewed distribution

right-skewed distribution

A right-skewed distribution has a peak to the left of the distribution and data values that taper off to the right.
shape

shape

The shape of a histogram can lead to valuable conclusions about the trend(s) of the data.
skewed

skewed

As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than or greater than the mode.
symmetric

symmetric

In statistics, a distribution is considered symmetric if  the data set that is mound-shaped.
symmetric histogram

symmetric histogram

For a symmetric histogram, the values of the mean, median, and mode are all the same and are all located at the center of the distribution.
undefined bimodal

undefined bimodal

A undefined bimodal histogram has a shape is not specifically defined, but we can note regardless that it is bimodal, having two separated classes or intervals equally representing the maximum frequency of the distribution.
uniform

uniform

A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.
unimodal

unimodal

If a data set has only 1 value that occurs most often, the set is called  unimodal.

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