Chapter 11

STA 200 Summer I 2011

Histograms

• Bar graphs and pie charts are appropriate graphs for categorical variables.

• To display the distribution of a quantitative variable graphically, we use a histogram.

• In most cases, quantitative variables take way too many values to have a separate bar for each value.

• Instead, we’ll have to group values together into intervals, or classes.

Creating a Histogram

• Divide the range of data into classes of equal width. Then, count the number of individuals in each class.

• When drawing the histogram:– the variable goes on the horizontal axis– the rate or count of occurrences goes on the vertical axis– the height of each bar is determined by the rate or count

of occurrences– the bars should touch

Example

• Suppose we have the following exam grades:

99 69 71 32 62

81 61 67 80 90

62 83 59 82 55

77 95 74 93 58

71 72 76 95 51

85 86 77 15 67

55 68 75 82 79

73 76 68 96 97

Example (cont.)

• Step 1: Choosing Classes– Make sure all of your classes are the same size.– Make sure your classes cover all of the data.– Make sure that none of your classes overlap.

• Here, we’re going to set up the classes in the most accessible manner:– 90-99– 80-89– 70-79– etc.

Example (cont.)

• Step 2: Counting

Class Count Class Count

90-99 40-49

80-89 30-39

70-79 20-29

60-69 10-19

50-59

Example (cont.)

Histogram:

Things to Look For

• Outliers:– observations outside the overall pattern of data– either significantly higher or significantly lower

than the rest of the data

• Shape– roughly symmetric, left-skewed, or right-skewed

Outliers

• In the exam score example, are there any outliers?

• In a histogram, the outliers will usually stand out:

02468

101214

Freq

uency

Series1

048

12

Freq

uency

Shape

• If a distribution is roughly symmetric, the left and right sides will be approximate mirror images of each other.

• If a distribution is skewed, one tail will be longer than the other.– left-skewed: long left tail– right-skewed: long right tail

More Shape

• Some types of data regularly produce distributions with a specific shape.

– Symmetric (not necessarily bell-shaped): the size of organisms of the same species, IQ scores

– Right-skewed: income

– Left-skewed: exam scores (usually)

Stem-and-Leaf Plots

• A quick, easy alternative to a histogram is a stem-and-leaf plot.

• For small data sets, a stem-and-leaf plot may be preferable to a histogram:– stem-and-leaf plots are quicker to make– they show more information than a histogram

Constructing a Stem-and-Leaf Plot• Separate each observation into a stem and a leaf.

– The leaf is the final digit.– The stem is everything but the final digit.

• Write the stems in a vertical column with the smallest one on top.

• Write each leaf in the row next to the appropriate stem. The leaves in each row should increase as you get farther away from the stem.

• If you have to round off the observations, be sure to use a key or legend to explain the rounding.

Example

• The following data represent the salaries (in millions) of a major league baseball team:

• Create a stem-and-leaf plot of the data.

3.0 4.0 0.4 2.6

0.8 0.4 1.5 4.1

0.4 7.0 5.0 1.5

2.0 2.9 15.0 0.9

0.4 0.5 4.0 5.1

3.2 0.4 9.4 0.4

1.6 12.5 0.8 0.6

Example (cont.)

Stem Leaf

What shape does the distribution have?