Frequency Distributions

Frequency Distributions

Frequency Distribution Table:

Shows classes (intervals, buckets) of data with the frequency (count) of the number of entries in each class.

Frequency Distributions

Class Frequency

1 - 5 3

6 - 10 5

11 - 15 4

16 - 20 2

21 - 25 1

Class Limits:Each class has an upper and lower limit. In this example: 1 is the lower class limit of the first class, and 5 is the upper class limit of the first class.

Frequency Distributions

Class Frequency

1 - 5 3

6 - 10 5

11 - 15 4

16 - 20 2

21 - 25 1

Class Width:The class width is the difference between the upper limits (or the lower limits) of adjacent classes.

In this example, the class width is 6 – 1 = 5

Frequency Distributions

Class Frequency

1 - 5 3

6 - 10 5

11 - 15 4

16 - 20 2

21 - 25 1

Class width6 - 1= 5

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Frequency Distributions

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Frequency Distributions

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Frequency Distributions

class tally f

67 – 78

79 - 90

91 - 102

103 - 114

115 - 126

Frequency distribution with 5 classes

Min: 67; Max: 125; Classes: 5

Class Width: (125 – 67)/5 = 11.6 → 12

Lower Class Limits:67, 79, 91, 103, 115

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Frequency Distributions

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Frequency Distributions

Class Midpoint: (lower limit + upper limit)/2

Notice the difference between the midpoints of consecutive classes is the class width.

84.5 – 72.5 = 1296.5 – 84.5 = 12

class midpoint f

67 – 78 (67 + 78)/2 = 72.5

3

79 - 90 (79+90)/2= 84.5

5

91 - 102 (91 + 102)/2= 96.5

8

103 - 114 (103 + 114)/2= 108.5

9

115 - 126 (115 + 126)/2= 120.5

5

30

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Frequency Distributions

Relative Frequency of a Class Portion of the whole in the class

Relative Frequency = f/n

class f Relative freq

67 – 78 3 3/30 = 1/10

79 - 90 5 5/30 =1/6

91 - 102 8 8/30 =4/15

103 - 114 9 9/30 =3/10

115 - 126 5 5/30

n = 30

30/30 = 1

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Frequency Distributions

Cumulative Frequency

The sum of the frequency of current class and all lower classes.

Think of it as a running total.

class f Cumulative Freq

67 – 78 3 3

79 - 90 5 8

91 - 102 8 16

103 - 114 9 25

115 - 126 5 30

n = 30

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Frequency Distributions

Expanded Frequency Distribution class f midpoint Relative freq Cumulative

freq

67 – 78 3 72.5 .10 3

79 - 90 5 84.5 .17 8

91 - 102 8 96.5 .27 16

103 - 114 9 108.5 .30 25

115 - 126 5 120.5 .17 30

n = 30

1.01 (rounding)

Midpoint = (lower limit + upper limit)/2

Relative Frequency = f/n

Cumulative Frequency:Number of values in that class or lower

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Frequency Distributions

Class Boundaries separate the data values in a class without being a data value in the class. Boundaries are usually set at one-half the precision of your data. For whole number values, we back off 0.5 in each direction:

class f Class Boundaries

67 – 78

3 66.5 – 78.5

79 - 90

5 78.5 – 90.5

91 - 102

8 90.5 – 102.5

103 - 114

8 102.5 – 114.5

115 - 126

5 114.5 – 126.5

n = 30

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Frequency Distributions

Class Boundaries

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Frequency Distributions

Class Boundaries

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Frequency Distributions

Frequency Histogram ● Bar graph of data classes on the horizontal axis vs frequency or relative frequency on the vertical

● The bars must touch

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Frequency Distributions

class limits

class midpoints

class boundaries

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Frequency Distributions

Frequency Histogramby class boundaries

This is the from most college texts will use.

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Frequency Distributions

By Class Limits

Online generators often use this form.

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Frequency Distributions

By Midpoints

Allows one to think of each bar as a count of a representative value for calculation purposes.

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Frequency Distributions

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Frequency Distributions

3

5

89

5

Typically the final product does not contain the bars.

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Frequency Distributions

Relative Frequency Histogram has the relative frequency as the

vertical axis.

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Frequency Distributions

Ogive Shows cumulative frequency at the upper class boundaries

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Frequency Distributions

Key: 6 | 7 = 67

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Frequency Distributions

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Frequency Distributions

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Data

Distributions of Data Sets: Shape

Some shapes of distributions are symmetric, uniform, skew right and skew left.

These are ideal models. With real data, you probably won’t get distributions that fit perfectly, but we will use these models as a general description.

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Data

Distributions of Data Sets: Shape

Shape of distributions: symmetric:

Also called bell shaped or nearly normal. The mean median,and mode are equal in a symmetric distribution.

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Data

Distributions of Data Sets: Shape

Shape of distribution: uniform

All entries or classes have equal frequencies. The mean And median are the same

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Data

Distributions of Data Sets: Shape

Shape of distribution: skew left

The “tail” is on the left. The mean is left of the median andThe mode.

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Data

Distributions of Data Sets: Shape

Shape of distribution: skew right

The “tail” is on the right. The mean is right of the median and the mode.

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Frequency Distributions