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The following data are the number of pages in 40 books on a shelf. Construct a box plot using a graphing calculator, and state the interquartile range.

  • 136
  • 140
  • 178
  • 190
  • 205
  • 215
  • 217
  • 218
  • 232
  • 234
  • 240
  • 255
  • 270
  • 275
  • 290
  • 301
  • 303
  • 315
  • 317
  • 318
  • 326
  • 333
  • 343
  • 349
  • 360
  • 369
  • 377
  • 388
  • 391
  • 392
  • 398
  • 400
  • 402
  • 405
  • 408
  • 422
  • 429
  • 450
  • 475
  • 512

IQR = 158

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For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same. For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like:

Horizontal boxplot box begins at the smallest value and Q1, 1, until the Q3 and median, 5, no median line is designated, and has its lone whisker extending from the Q3 to the largest value, 7.

In this case, at least 25% of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25% of the values are equal to five. The top 25% of the values fall between five and seven, inclusive.

Test scores for a college statistics class held during the day are:

  • 99
  • 56
  • 78
  • 55.5
  • 32
  • 90
  • 80
  • 81
  • 56
  • 59
  • 45
  • 77
  • 84.5
  • 84
  • 70
  • 72
  • 68
  • 32
  • 79
  • 90

Test scores for a college statistics class held during the evening are:

  • 98
  • 78
  • 68
  • 83
  • 81
  • 89
  • 88
  • 76
  • 65
  • 45
  • 98
  • 90
  • 80
  • 84.5
  • 85
  • 79
  • 78
  • 98
  • 90
  • 79
  • 81
  • 25.5

  1. Find the smallest and largest values, the median, and the first and third quartile for the day class.
  2. Find the smallest and largest values, the median, and the first and third quartile for the night class.
  3. For each data set, what percentage of the data is between the smallest value and the first quartile? the first quartile and the median? the median and the third quartile? the third quartile and the largest value? What percentage of the data is between the first quartile and the largest value?
  4. Create a box plot for each set of data. Use one number line for both box plots.
  5. Which box plot has the widest spread for the middle 50% of the data (the data between the first and third quartiles)? What does this mean for that set of data in comparison to the other set of data?
    • Min = 32
    • Q 1 = 56
    • M = 74.5
    • Q 3 = 82.5
    • Max = 99
    • Min = 25.5
    • Q 1 = 78
    • M = 81
    • Q 3 = 89
    • Max = 98
  1. Day class: There are six data values ranging from 32 to 56: 30%. There are six data values ranging from 56 to 74.5: 30%. There are five data values ranging from 74.5 to 82.5: 25%. There are five data values ranging from 82.5 to 99: 25%. There are 16 data values between the first quartile, 56, and the largest value, 99: 75%. Night class:
  2. Two box plots over a number line from 0 to 100.  The top plot shows a whisker from 32 to 56, a solid line at 56, a dashed line at 74.5, a solid line at 82.5, and a whisker from 82.5 to 99.  The lower plot shows a whisker from 25.5 to 78, solid line at 78, dashed line at 81, solid line at 89, and a whisker from 89 to 98.
  3. The first data set has the wider spread for the middle 50% of the data. The IQR for the first data set is greater than the IQR for the second set. This means that there is more variability in the middle 50% of the first data set.
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The following data set shows the heights in inches for the boys in a class of 40 students.

66; 66; 67; 67; 68; 68; 68; 68; 68; 69; 69; 69; 70; 71; 72; 72; 72; 73; 73; 74
The following data set shows the heights in inches for the girls in a class of 40 students.
61; 61; 62; 62; 63; 63; 63; 65; 65; 65; 66; 66; 66; 67; 68; 68; 68; 69; 69; 69
Construct a box plot using a graphing calculator for each data set, and state which box plot has the wider spread for the middle 50% of the data.

IQR for the boys = 4

IQR for the girls = 5

The box plot for the heights of the girls has the wider spread for the middle 50% of the data.

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Graph a box-and-whisker plot for the data values shown.

  • 10
  • 10
  • 10
  • 15
  • 35
  • 75
  • 90
  • 95
  • 100
  • 175
  • 420
  • 490
  • 515
  • 515
  • 790

The five numbers used to create a box-and-whisker plot are:

  • Min: 10
  • Q 1 : 15
  • Med: 95
  • Q 3 : 490
  • Max: 790

The following graph shows the box-and-whisker plot.

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Follow the steps you used to graph a box-and-whisker plot for the data values shown.

  • 0
  • 5
  • 5
  • 15
  • 30
  • 30
  • 45
  • 50
  • 50
  • 60
  • 75
  • 110
  • 140
  • 240
  • 330

The data are in order from least to greatest. There are 15 values, so the eighth number in order is the median: 50. There are seven data values written to the left of the median and 7 values to the right. The five values that are used to create the boxplot are:

  • Min: 0
  • Q 1 : 15
  • Med: 50
  • Q 3 : 110
  • Max: 330
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References

Data from West Magazine .

Chapter review

Box plots are a type of graph that can help visually organize data. To graph a box plot the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data.

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.

Construct a box plot below. Use a ruler to measure and scale accurately.

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Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas, but not in others? How can you tell?

More than 25% of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25% and the bottom 25% are spread out evenly; the whiskers have the same length.

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Bringing it together

Santa Clara County, CA, has approximately 27,873 Japanese-Americans. Their ages are as follows:

Age Group Percent of Community
0–17 18.9
18–24 8.0
25–34 22.8
35–44 15.0
45–54 13.1
55–64 11.9
65+ 10.3
  1. Construct a histogram of the Japanese-American community in Santa Clara County, CA. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph?
  2. What percentage of the community is under age 35?
  3. Which box plot most resembles the information above?
Three box plots with values between 0 and 100.  Plot i has Q1 at 24, M at 34, and Q3 at 53; Plot ii has Q1 at 18, M at 34, and Q3 at 45; Plot iii has Q1 at 24, M at 25, and Q3 at 54.
  1. For graph, check student's solution.
  2. 49.7% of the community is under the age of 35.
  3. Based on the information in the table, graph (a) most closely represents the data.
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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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