MATH225 Week 2 Assignment Frequency Tables and Histograms (July 2019)

MATH 225 Statistical Reasoning for the Health Sciences

Week 2 Assignment Frequency Tables and Histograms

Question The histogram below represents the prices of digital SLR camera models at a store. Describe the shape of the distribution.

A histogram has a horizontal axis labeled Camera Prices in dollars from 0 to 2000 in increments of 500 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 250 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 5; 250, 7; 500, 4; 750, 3; 1000, 3; 1250, 2; 1500, 1; 1750, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 0; 2, 0; 3, 0; 4, 1; 5, 2; 6, 2; 7, 4; 8, 6; 9, 7; 10, 6; 11, 5; 12, 3; 13, 2; 14, 1; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A restaurant is open for both lunch and dinner. One day, the owner kept track of the number of occupied tables in the dining area and created a histogram showing the results for each half hour of the day. What shape does the distribution have?

A histogram has a horizontal axis labeled Time of Day from 9 to 21 in increments of 3 and a vertical axis labeled Frequency from 0 to 20 in increments of 5. The histogram contains vertical bars of width 0.5 starting at the horizontal axis value 9. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 9, 1; 9.5, 1; 10, 2; 10, 5; 11, 9; 11.5, 10; 12, 11; 12.5, 10; 13, 7; 13.5, 3; 14, 3; 14.5, 4; 15, 4; 15.5, 6; 16, 4; 16.5, 6; 17, 7; 17.5, 10; 18, 12; 18.5, 15; 19, 12; 19.5, 10; 20, 7; 20.5, 6. All vertical coordinates are approximate.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A bookstore took an inventory of the prices of its books and created a histogram to show the results. What shape does the distribution have?

A histogram has a horizontal axis labeled Book Prices in dollars from 0 to 200 in increments of 50 and a vertical axis labeled Frequency from 0 to 30 in increments of 10. The histogram contains vertical bars of width 10 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 5; 10, 20; 20, 25; 30, 21; 40, 15; 50, 13; 60, 14; 70, 16; 80, 16; 90, 21; 100, 24; 110, 26; 120, 22; 130, 21; 140, 20; 150, 15; 160, 9; 170, 7; 180, 5; 190, 3. All vertical coordinates are approximate.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 0; 2, 0; 3, 0; 4, 1; 5, 1; 6, 2; 7, 2; 8, 3; 9, 4; 10, 6; 11, 7; 12, 8; 13, 8; 14, 5; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 1; 2, 1; 3, 2; 4, 3; 5, 5; 6, 6; 7, 6; 8, 5; 9, 4; 10, 2; 11, 1; 12, 1; 13, 0; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left skewed

unimodal and right skewed

bimodal

Identify and Labels Shapes of Histograms

Histogram Shapes

A histogram is a graph that helps show the distribution of values in a set of data. An example of a histogram is shown below. The horizontal axis is labeled with data values. It is divided into several sections that all have the same width. Then a bar is drawn above each section, and the height of the bar is related to how many of the data values are within the corresponding range on the horizontal axis. The vertical axis (and the height of the bars) can be either counts of data values (called a frequency) or the fraction of the data set in the range (called the relative frequency).

A histogram has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

The general shape of a histogram can be described as uniform, unimodal, bimodal, or multimodal. An example of each of these is given below. A uniform histogram has bars that are all close to the same height. A unimodal histogram has a single peak, and a multimodal histogram has more one peak. Sometimes the case with two peaks is also called bimodal.

A histogram has a horizontal axis from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 6; 1, 6; 2, 6; 3, 6; 4, 6; 5, 6; 6, 6; 7, 6; 8, 6; 9, 6.

Uniform

A histogram has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

Unimodal

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 4; 9, 6; 10, 7; 11, 7; 12, 6; 13, 4; 14, 2; 15, 1.

Bimodal

A histogram has a horizontal axis from 0 to 20 in increments of 4 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 5; 4, 4; 5, 3; 6, 4; 7, 6; 8, 7; 9, 8; 10, 7; 11, 6; 12, 5; 13, 4; 14, 4; 15, 5; 16, 6; 17, 4; 18, 2; 19, 1.

Multimodal

A histogram can also be described by it symmetry or skewness. A symmetric histogram has two halves that are approximately mirror images of each other. The uniform, unimodal, and bimodal histograms shown above are all symmetric. A histogram is skewed left when the tail of bars extending towards smaller data values is longer than the tail extending towards larger data values. A histogram is skewed right when the tail of bars extending towards larger data values is longer than the tail extending towards smaller data values. Examples of skewed histograms are shown below.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 1; 2, 1; 3, 1; 4, 2; 5, 2; 6, 3; 7, 3; 8, 4; 9, 5; 10, 6; 11, 7; 12, 7; 13, 5; 14, 3; 15, 1.

Skewed Left

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 3; 2, 5; 3, 7; 4, 7; 5, 6; 6, 5; 7, 4; 8, 3; 9, 3; 10, 2; 11, 2; 12, 1; 13, 1; 14, 1; 15, 0.

Skewed Right

Real world distributions tend to be right skewed when there is a lower bound for the values, most values are clustered in a range, but it is not impossible for very large values to occur. A classic example of this is income distribution is some countries. The lower bound is zero, most of the population have incomes within a few standard deviations of the mean, but there are people with much larger incomes.

Similarly, left skewed distributions tend to occur for quantities that have a natural upper bound when most of the population tend to be near that bound. For example, student scores on a easy test will tend to all be fairly high and create a peak near 100%. However, it is possible for a few students to create a tail that extends down to much lower scores.

Uniform distributions are less common. One simple example is rolling a fair dice. Every number from 1to 6 has the same probability of appearing. Bimodal distributions occur when there is a reason for two different peaks. For example, the distribution of the people attending Disney Land at different timings could be bimodal. It could have a peak at 11 am and another peak at 2 pm.

The unimodal symmetric distribution, also called bell shaped is a very common distribution. For example, the distribution of the mean wages of many random samples of 30 people will be a unimodal symmetric distribution.

Example

Question Describe the shape of the histogram shown below.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 2; 1, 7; 2, 8; 3, 7; 4, 6; 5, 4; 6, 3; 7, 3; 8, 2; 9, 2; 10, 1; 11, 1; 12, 1; 13, 0; 14, 0; 15, 0.

Question A professor created a histogram showing the birth month of the students in one of her classes. What is the shape of the histogram?

A histogram has a horizontal axis labeled Month of Birth from 1 to 12 with the following tick marks from left to right: 1, 3, 6, 9, and 12. It has a vertical axis labeled Frequency from 0 to 10 in increments of 2. Vertical bars of width 1 start at the horizontal axis value 1. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 1, 7; 2, 6; 3, 7; 4, 6; 5, 6; 6, 5; 7, 6; 8, 6; 9, 7; 10, 6; 11, 6.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A student in a probability class rolled a six-sided die 1000 times. A histogram of the results is shown below. What is the shape of the distribution?

A histogram has a horizontal axis labeled Die Roll from 1 to 6 in increments of 1 and a vertical axis labeled Frequency from 0 to 200 in increments of 50. Vertical bars of width 1 are centered over a horizontal axis label. The heights of the bars are as follows, where the horizontal axis label is listed first and the approximate height is listed second: 1, 170; 2, 150; 3, 155; 4, 150; 5, 170; 6, 165.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Given the following histogram for a set of data, how many values in the data set are at least 5.5 and less than 8.5?

A histogram has a horizontal axis labeled Values from 3.5 to 10.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram has vertical bars of width 1, starting at the horizontal axis value of 3.5. The approximate heights of the bars are as follows, where the horizontal axis label is listed first and the approximate height is listed second: 3.5, 5; 4.5, 6; 5.5, 7; 6.5, 5; 7.5, 5; 8.5, 5; 9.5, 6.

Question The students in a statistics class record how many movies they have watched in the previous month. The data are listed below.

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

Which of the histograms below correctly represents this data?

A histogram has a horizontal axis labeled Movies Watched from 0.5 to 6.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 8 in increments of 1. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 0.5, 2; 1.5, 8; 2.5, 3; 3.5, 2; 4.5, 1; 5.5, 1.

A histogram has a horizontal axis labeled Movies Watched from 0.5 to 6.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 0.5, 3; 1.5, 7; 2.5, 2; 3.5, 3; 4.5, 1; 5.5, 1.

A histogram has a horizontal axis labeled Movies Watched from 0.5 to 6.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 0.5, 3; 1.5, 7; 2.5, 3; 3.5, 2; 4.5, 1; 5.5, 1.

A bar graph has a horizontal axis titled Values labeled from 0.5 to 6.5 in increments of 1 and a vertical axis titled Frequency labeled from 1 to 6 in increments of 1. Six bars are plotted each with a width of 1. From left to right, the heights of the bars are as follows: 2, 6, 3, 2, 1, 1.

A bar graph has a horizontal axis titled Movies Watched labeled from 0.5 to 6.5 in increments of 1 and a vertical axis titled Frequency labeled from 1 to 8 in increments of 1. Six bars are plotted each with a width of 1. From left to right, the heights of the bars are as follows: 3, 8, 3, 1, 2, 1.

Histograms

Constructing Histograms

To construct a histogram,

1. Decide how many bars or intervals you need to clearly represent the data. (On average, most histograms consist of 5 to 15 bars.)

2. Choose a starting point for the first interval. This value should be less than the smallest data value. It is helpful to choose a starting point that is also carried out to one more decimal placethan the data value with the most decimal places. For example, if the value with the most decimal places is 6.1, and this is the smallest value, a good starting point is 6.05 (6.1 – 0.05 = 6.05).

3. Choose an ending point for the last interval. This value should be greater than the highest data value. Like the starting point, It is helpful to choose an ending point that is also carried out to one more decimal place than the data value with the most decimal places.

*Note: When these points and other boundaries are carried to one additional decimal place, no data value will fall on a boundary.

4. Calculate the width of the each bar or intervals. All intervals will be the same size. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (the number of bars you chose).

5. Determine the boundaries by adding the width to the starting point. Then add the width to that value, and continue as such. Label the boundary values on the horizontal axis.

6. Draw bars in each interval with the height corresponding to the frequency of data values that lie within each interval.

Example

Question Use the following data to construct a histogram.

The following data are the number of books bought by 50 part-time college students at ABC College.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

2, 2, 2, 2, 2, 2, 2, 2, 2, 2,

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,

4, 4, 4, 4, 4, 4,

5, 5, 5, 5, 5,

6, 6

Eleven students buy 1 book. Ten students buy 2 books. Sixteen students buy 3 books. Six students buy 4 books. Five students buy 5 books. Two students buy 6 books.

Question Given the following histogram for a set of data, how many values in the data set are greater than 6.5and less than 9.5?

A histogram has a horizontal axis labeled Values from 5.5 to 12.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 14 in increments of 2. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 5.5, 5; 6.5, 3; 7.5, 6; 8.5, 5; 9.5, 4; 10.5, 3; 11.5, 13.

9

10

13

14

18

Histograms

Interpreting Histograms

A histogram is a graph that consists of contiguous (adjoining) boxes, and can show you the shape, the center and the spread of the data. One advantage of a histogram is that it can readily display large data sets. A histogram has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with either the frequency or the relative frequency.

A histogram has a horizontal axis labeled Number of books from 0.5 to 6.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 16 in increments of 2. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 0.5, 11; 1.5, 10; 2.5, 16; 3.5, 6; 4.5, 5; 5.5, 2.

For example, in the histogram above, the horizontal axis represents the average number of books read by Mr. Rucker’s students each month. The vertical axis represents the number of students who read the corresponding number of books. The tallest bar (3rd from the left) in the histogram represents the number of students who read between 2.5 – 3.5 books on average each month. (These values can be found on the horizontal axis.) The height of this bar is 16 (found on the vertical axis. This means that 16 students read between 2.5 – 3.5 books on average each month.

Example

Question Given the histogram above, how many students read between 4.5 – 5.5 books on average each month?

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 5; 4, 4; 5, 2; 6, 1; 7, 1; 8, 2; 9, 4; 10, 6; 11, 7; 12, 8; 13, 8; 14, 6; 15, 2.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A student surveys his class and creates a histogram showing the number of pets in each student’s house. What is the shape of the distribution?

A histogram has a horizontal axis labeled Book Price in dollars from 0 to 4 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. Vertical bars of width 1 are centered over a horizontal axis label. The heights of the bars are as follows, where the horizontal axis label is listed first and the height is listed second: 0, 5; 1, 6; 2, 3; 3, 2; 4, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 1; 2, 1; 3, 2; 4, 3; 5, 5; 6, 6; 7, 6; 8, 5; 9, 4; 10, 2; 11, 1; 12, 1; 13, 0; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left skewed

unimodal and right skewed

bimodal

Identify and Labels Shapes of Histograms

Histogram Shapes

A histogram is a graph that helps show the distribution of values in a set of data. An example of a histogram is shown below. The horizontal axis is labeled with data values. It is divided into several sections that all have the same width. Then a bar is drawn above each section, and the height of the bar is related to how many of the data values are within the corresponding range on the horizontal axis. The vertical axis (and the height of the bars) can be either counts of data values (called a frequency) or the fraction of the data set in the range (called the relative frequency).

A histogram has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

The general shape of a histogram can be described as uniform, unimodal, bimodal, or multimodal. An example of each of these is given below. A uniform histogram has bars that are all close to the same height. A unimodal histogram has a single peak, and a multimodal histogram has more one peak. Sometimes the case with two peaks is also called bimodal.

A histogram has a horizontal axis from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 6; 1, 6; 2, 6; 3, 6; 4, 6; 5, 6; 6, 6; 7, 6; 8, 6; 9, 6.

Uniform

A histogram has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

Unimodal

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 4; 9, 6; 10, 7; 11, 7; 12, 6; 13, 4; 14, 2; 15, 1.

Bimodal

A histogram has a horizontal axis from 0 to 20 in increments of 4 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 5; 4, 4; 5, 3; 6, 4; 7, 6; 8, 7; 9, 8; 10, 7; 11, 6; 12, 5; 13, 4; 14, 4; 15, 5; 16, 6; 17, 4; 18, 2; 19, 1.

Multimodal

A histogram can also be described by it symmetry or skewness. A symmetric histogram has two halves that are approximately mirror images of each other. The uniform, unimodal, and bimodal histograms shown above are all symmetric. A histogram is skewed left when the tail of bars extending towards smaller data values is longer than the tail extending towards larger data values. A histogram is skewed right when the tail of bars extending towards larger data values is longer than the tail extending towards smaller data values. Examples of skewed histograms are shown below.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 1; 2, 1; 3, 1; 4, 2; 5, 2; 6, 3; 7, 3; 8, 4; 9, 5; 10, 6; 11, 7; 12, 7; 13, 5; 14, 3; 15, 1.

Skewed Left

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 1; 1, 3; 2, 5; 3, 7; 4, 7; 5, 6; 6, 5; 7, 4; 8, 3; 9, 3; 10, 2; 11, 2; 12, 1; 13, 1; 14, 1; 15, 0.

Skewed Right

Real world distributions tend to be right skewed when there is a lower bound for the values, most values are clustered in a range, but it is not impossible for very large values to occur. A classic example of this is income distribution is some countries. The lower bound is zero, most of the population have incomes within a few standard deviations of the mean, but there are people with much larger incomes.

Similarly, left skewed distributions tend to occur for quantities that have a natural upper bound when most of the population tend to be near that bound. For example, student scores on a easy test will tend to all be fairly high and create a peak near 100%. However, it is possible for a few students to create a tail that extends down to much lower scores.

Uniform distributions are less common. One simple example is rolling a fair dice. Every number from 1to 6 has the same probability of appearing. Bimodal distributions occur when there is a reason for two different peaks. For example, the distribution of the people attending Disney Land at different timings could be bimodal. It could have a peak at 11 am and another peak at 2 pm.

The unimodal symmetric distribution, also called bell shaped is a very common distribution. For example, the distribution of the mean wages of many random samples of 30 people will be a unimodal symmetric distribution.

Example

Question Describe the shape of the histogram shown below.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 2; 1, 7; 2, 8; 3, 7; 4, 6; 5, 4; 6, 3; 7, 3; 8, 2; 9, 2; 10, 1; 11, 1; 12, 1; 13, 0; 14, 0; 15, 0.

Question The histogram shows the income of the families of the students in a statistics class. What is the shape of the histogram?

A histogram has a horizontal axis labeled Income in thousands from 0 to 200 in increments of 40 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. Vertical bars of width 20 start at the horizontal axis value 0. The heights of the bars are as follows, where the left horizontal axis label is listed first and the frequency is listed second: 0, 6; 20, 8; 40, 9; 60, 8; 80, 6; 100, 4; 120, 3; 140, 2; 160, 1; 180, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question The kindergarten students in a school were asked to reach into a bag of candy and pull out as many pieces as they could with one hand. The number of candies for each student was counted, and the results are displayed in the following frequency table.

Which histogram accurately summarizes the data?

Value

Frequency

8

2

9

6

10

2

11

4

12

6

13

7

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A histogram has a horizontal axis labeled Values from 5.5 to 11.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 5.5, 2; 6.5, 6; 7.5, 2; 8.5, 4; 9.5, 6; 10.5, 7.

A histogram has a horizontal axis labeled Values from 9.5 to 15.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 9.5, 2; 10.5, 6; 11.5, 2; 12.5, 4; 13.5, 6; 14.5, 7.

A bar graph has a horizontal axis titled Values labeled from 7.5 to 13.5 in increments of 1 and a vertical axis titled Frequency labeled from 0 to 7 in increments of 1. Six bars are plotted each with a width of 1. From left to right, the heights of the bars are as follows: 2, 6, 2, 4, 6, 7.

A bar graph has a horizontal axis titled Values labeled from 3.5 to 9.5 in increments of 1 and a vertical axis titled Frequency labeled from 0 to 7 in increments of 1. Six bars are plotted each with a width of 1. From left to right, the heights of the bars are as follows: 2, 6, 2, 4, 6, 7.

Question Several people were asked to report the number of hours of sleep they average per night. The results are shown in the histogram below. How many of those people average greater than 4.5 and less than 6.5 hours of sleep per night?

A histogram has a horizontal axis labeled Values from 3.5 to 8.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 12 in increments of 2. The histogram contains vertical bars of width 1, with one vertical bar centered over each of the horizontal axis value tick marks. The heights of the vertical bars are as follows, where the value is listed first and the height is listed second: 3.5, 6; 4.5, 7; 5.5, 4; 6.5, 5; 7.5, 11.

Question Several executives were asked how many suits they own. The results are tabulated in the following frequency table.

Which histogram accurately summarizes the data?

Value

Frequency

8

6

9

5

10

3

11

5

12

3

13

2

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A histogram has a horizontal axis labeled Values from 7.5 to 13.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 6 in increments of 1. The histogram has vertical bars of width 1, starting at the horizontal axis value of 7.5. The approximate heights of the bars are as follows, where the horizontal axis label is listed first and the approximate height is listed second: 7.5, 6; 8.5, 5; 9.5, 3; 10.5, 5; 11.5, 3; 12.5, 2.

A histogram has a horizontal axis labeled Values from 11.5 to 17.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 6 in increments of 1. The histogram has vertical bars of width 1, starting at the horizontal axis value of 11.5. The approximate heights of the bars are as follows, where the horizontal axis label is listed first and the approximate height is listed second: 11.5, 6; 12.5, 5; 13.5, 3; 14.5, 5; 15.5, 3; 16.5, 2.

A histogram has a horizontal axis labeled Values from 9.5 to 15.5 in increments of 1 and a vertical axis labeled Frequency from 0 to 6 in increments of 1. The histogram has vertical bars of width 1, starting at the horizontal axis value of 9.5. The approximate heights of the bars are as follows, where the horizontal axis label is listed first and the approximate height is listed second: 9.5, 6; 10.5, 5; 11.5, 3; 12.5, 5; 13.5, 3; 14.5, 2.

A bar graph has a horizontal axis titled Values labeled from 5.5 to 11.5 in increments of 1 and a vertical axis titled Frequency labeled from 0 to 6 in increments of 1. Six bars are plotted each with a width of 1. From left to right, the heights of the bars are as follows: 6,5,3,5,3,2.

Question Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 0; 1, 0; 2, 0; 3, 0; 4, 4; 5, 5; 6, 5; 7, 5; 8, 5; 9, 4; 10, 5; 11, 5; 12, 6; 13, 5; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question

Describe the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains vertical bars of width 1 starting at the horizontal axis value 0. The heights of the bars are as follows, where the horizontal axis label is listed first and the frequency is listed second: 0, 7; 1, 8; 2, 8; 3, 7; 4, 6; 5, 4; 6, 4; 7, 3; 8, 2; 9, 2; 10, 1; 11, 1; 12, 0; 13, 0; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal