Histograms

Lecture 14Sec. 4.4.4Fri, Feb 8, 2008

Histograms Histogram Classes

Histograms vs. Bar Graphs

Bar graphs are for qualitative data Histograms are for quantitative data. We indicate this difference by leaving a

gap between the bars of a bar graph and no gap between the rectangles of a histogram.

Example

Draw a histogram of the following data.

2.946 2.335 3.418 1.890

2.731 3.855 1.344 2.126

2.881 2.542 2.504 3.367

1.950 2.392 2.443 3.053

Drawing Histograms Find the maximum value, the minimum

value, and the range.Minimum = 1.344Maximum = 3.855Range = Max – Min = 3.855 – 1.344 = 2.511.

Drawing Histograms Divide the data into classes of equal width. The classes must not overlap.

Choose a convenient starting point.Choose a convenient class width. Write the endpoints of each class.

Drawing Histograms Let’s use 6 classes Then the width must be at least 2.511/6 =

0.4185. Let’s use 0.5 (other choices are possible). Starting point = 1.0 (other choices are

possible).

Drawing Histograms Or we could begin by choosing the class

width. Let the class width be 0.5 (other choices

are possible). Then the number of classes will be at least

2.511/0.5 = 5.022, or 6. Starting point = 1.0.

Drawing Histograms Classes:

1.0 up to 1.5 (but not including 1.5)1.5 up to 2.02.0 up to 2.52.5 up to 3.03.0 up to 3.53.5 up to 4.0

Drawing Histograms We may write the classes in either of two ways. Interval notation: [low, high)

[1.0, 1.5), [1.5, 2.0), [2.0, 2.5), etc.

[ and ] mean “include endpoints.” ( and ) mean “exclude endpoints.”

Drawing Histograms Range notation: low – high

1.000 – 1.499, 1.500 – 1.999, 2.000 – 2.499, etc.

With this notation, the endpoints are assumed to be included.

Therefore, be sure the endpoints do not overlap. Yet be sure that no possible values are left out.

Drawing Histograms Count the number of observations in each

class. This is the frequency of the class.

Class Frequency

1.000 – 1.499 1

1.499 – 1.999 2

2.000 – 2.499 4

2.500 – 2.999 5

3.000 – 3.499 3

3.500 – 3.999 1

Drawing Histograms Draw horizontal and vertical axes. On the horizontal axis, show the class

limits. On the vertical axis, show uniform

reference points representing frequencies or precentages that are appropriate for the data, starting at 0.

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5

GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms Over each class, draw a rectangle whose

height is the frequency, or relative frequency, of that class.

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms

1.0 1.5 2.0 2.5 3.0 3.5GPA

2

4

6

8

0

Frequency

4.0

Drawing Histograms Never use too few or too many classes.

Usually 5 to 12 classes is about right.

Use simple round numbers for the class boundaries.

Mark off the vertical axis uniformly, showing regular reference points, not the actual frequencies.

The vertical scale must start at 0.

TI-83 – Histograms Enter the data into list L1.

{2.946, 2.731, 2.881, …, 3.053} L1

Press STAT PLOTSelect Plot1.Press Enter.Turn Plot1 On.Select Histogram Type.Specify List L1.

TI-83 – Histograms Press WINDOW

Set Xmin to the starting point.Set Xmax to the last endpoint.Set Xscl to the class width.Set Ymin to 0 (or -1 for a margin).Set Ymax to the maximum frequency.

Press GRAPHThe histogram appears.

TI-83 – Histograms Or press ZOOM

Select ZoomStat (#9).The histogram appears.

TI-83 – Frequency Distributions

After getting the histogram, press TRACE.The display shows the first class and its

frequency.Use the left arrow to see the other class

frequencies.