Cummulative Frequency Distribution

8/8/2019 Cummulative Frequency Distribution

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CUMMULATIVE FREQUENCY DISTRIBUTION

I. Ranked Distributions

Frequency distributions organize Raw Data or numbers that have been

collected. The first step in the process of organizing your newly collected raw data is togenerate a Ranked Distribution. Ranked distributions simply rank order all of thenumbers of your raw data.

The following are examples scores for 20 students on the first examination:

51 98 55 71 87 82 83 55 90 65 76 90 71 82 97 67 99 71 8859

To create a Ranked Distribution rearrange the data from the highest numberto the lowest.

The following are the same scores as above but in a ranked distribution:

99 98 97 90 90 88 87 83 82 82 76 71 71 71 67 65 59 55 55 51

II. Frequency Distributions

Simple Frequency Distributions are created by listing all the possible score

values in any distribution and then indicating the frequency (how often each scoreoccurs). Frequency Distributions are useful only if they simplify the data. The tablebelow shows the raw data from the above example in a Frequency Distribution:

Grade Score Frequency

99 1

98 1

87 1

A 90 2

88 1

87 183 1

B 82 2

76 1

C 71 3

67 1

D 65 1


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F 59 1

55 2

51 1

Total 20

Notice that there are 2 scores of 90 and 3 scores of 71 in the original data setabove. These values are represented as a frequency of 2 and 3 respectively in theFrequency Distribution. Notice also that the total frequency at the bottom (20) is thesame number as the number of raw data data points you have.

III. Grouped Frequency Distributions

When you have a high amount of unique scores you should generate a

Grouped Frequency Distribution. In a grouped frequency distribution raw data arecombined into equalized groups called class intervals. The grouped frequencydistributiongives you the whole picture at a glance.

Grade Group Frequency

A 90-99 5

B 80-89 5

C 70-79 4

D 60-69 2

F 50-59 4

Total 20

Constructing the Class Intervals

Construction of your class intervals is largely dependent on the type of datayou are working with. When dealing with grade data as above then class interval sizesof 10 and a total of 5 intervals works best. For most data there are several differentways that you could construct your class intervals and no one is necessarily better thananother. There are some general rules about class intervals that make the data easierto understand. Good interval size numbers are multiples of 2, 5, or 10. Generally

speaking, interval sizes should be between 10 and 20. Less than 10 results in loss ofinformation about the original data and more than 20 is difficult to comprehend. Thenumber of class intervals should reside somewhere between 5 and 20. You cancalculate the number of intervals and interval size that would be best for any set ofdata.


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To determine the number of intervals needed you first need to compute therange of your data:

Range = high score low scoreFor our original data:

= 99 51Range = 48

The second step is to select an interval size (i). Let's say you select aninterval size of 5. Use the formula below to calculate the number of intervals you shoulduse.

number of intervals ~ range/i (interval size)~ 48/5~ 9.6 or rounded to 10

so we can use 10 intervals with a class interval size of 5 to represent our data:

Interval Frequency

95-99 3

90-94 2

85-89 2

80-74 3

75-79 1

70-74 365-69 2

60-64 0

55-59 3

50-54 1

Total 20

Each class interval is represented by a lower limit (e.g., 95 for the top interval) and anupper limit (e.g., 99 for the top interval). It is usually best to establish a lower limit that

is a multiple of the interval size. this makes the table easier to understand. Once theintervals are complete you simply count the number of data points (or frequency) thatfit within each class interval.

To calculate the interval size (i) that would be best for any set of data youfirst need to compute the range of your data:


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Range = high score low scoreFor our original data:

= 99 51Range = 48

The second step is to select the number of intervals you would like to use.Let's say you select 10 intervals. Use the formula below to calculate the interval sizeyou should use.

i (interval size) ~ range/number of intervals~ 48/10~ 4.8 or rounded to 5

IV. Apparent Limits and Real Limits

Apparent Limits are the same units as the original data while Real Limits arethe lower apparent limit minus 0.5 and the upper apparent limit plus 0.5. Notice thedifference in the table below:

Real LimitsApparent

LimitsFrequency

94.5-99.5 95-99 3

89.5-94.5 90-94 2

84.5-89.5 85-89 2

79.5-74.5 80-74 3

74.5-79.5 75-79 1

69.5-74.5 70-74 3

64.5-69.5 65-69 2

59.5-64.5 60-64 0

54.5-59.5 55-59 3

49.5-54.5 50-54 1

Total 20


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V. Midpoint

The Midpoint is the exact center of an interval. When the interval size is oddthe midpoints will be whole numbers. When the interval size is even the midpoints will

end in .5. The midpoint is calculated with the formula below:

Midpoint = lower limit + upper limit2

RealLimits

ApparentLimits

MidpointsFrequency

94.5-99.5

95-99 97 3

89.5-

94.5

90-94 92 2

84.5-89.5

85-89 87 2

79.5-74.5

80-74 82 3

74.5-79.5

75-79 77 1

69.5-74.5

70-74 72 3

64.5-69.5

65-69 67 2

59.5-64.5

60-64 62 0

54.5-59.5

55-59 57 3

49.5-54.5

50-54 52 1

Total 20


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VI. Cumulative Frequency

IfFrequency is the total number of scores that fall within a class interval, thenCumulative Frequency is the total number of scores that fall below the upper reallimit of an interval. This is useful when you need to know how many scores fall below a

particular score. The easiest way to calculate cumulative frequency is to start at thebottom interval and add the Frequency scores as you move up the table. Thistechnique and the final outcome are shown in the table below:

RealLimits

ApparentLimits

MidpointsFrequencyCalculationCumulativeFrequency

94.5-99.5

95-99 97 3 17 + 3 = 20*

89.5-94.5

90-94 92 2 15 + 2 = 17

84.5-89.5 85-89 87 2 13 + 2 = 15

79.5-74.5

80-74 82 3 10 + 3 = 13

74.5-79.5

75-79 77 1 9 + 1 = 10

69.5-74.5

70-74 72 3 6 + 3 = 9

64.5-69.5

65-69 67 2 4 + 2 = 6

59.5-

64.5 60-64 62 0 4 + 0 = 4

54.5-59.5

55-59 57 3 1 + 3 = 4

49.5-54.5

50-54 52 1 1 = 1

Total 20*

*Note that the final cumulative frequency score should equal the total

frequency score.


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VII. Relative Frequency

Relative Frequency is used if you want to compare the frequencies of onedistribution with another when the total number of data points is different. Relative

Frequency is the proportion of scores from the distribution that fall within the reallimits of an interval This is similar to a percentage of scores where the percentage is theproportion multiplied by 100. The Relative Frequency is computed by dividing thefrequency in the interval by the total frequency or total number of scores (n):

Relative Frequency = frequency/ n

RealLimits

ApparentLimits

MidpointsFrequencyCumulativeFrequency

RelativeFrequency

94.5-99.5 95-99 97 3 20 0.15

89.5-94.5 90-94 92 2 17 0.10

84.5-89.5 85-89 87 2 15 0.10

79.5-74.5 80-74 82 3 13 0.15

74.5-79.5 75-79 77 1 10 0.05

69.5-74.5 70-74 72 3 9 0.15

64.5-69.5 65-69 67 2 6 0.10

59.5-64.5 60-64 62 0 4 0.00

54.5-59.5 55-59 57 3 4 0.15

49.5-54.5 50-54 52 1 1 0.05

Total 20 1.00*

*Note that the sum of the relative frequency should equal 1.00.


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VIII. Cumulative Relative Frequency

Cumulative Relative Frequency is the total proportion of scores that liebelow the real upper limit of the interval.

The easiest way to calculate cumulative relative frequency is to start at thebottom interval and add the Relative Frequency scores as you move up the table aswe did with Cumulative Frequency. This final outcome is shown in the table below:

RealLimits

ApparentLimits

MidpointsFrequencyCumulativeFrequency

RelativeFrequency

CumulativeRelative

Frequency

94.5-99.5 95-99 97 3 20 0.15 1.00*89.5-94.5 90-94 92 2 17 0.10 0.85

84.5-89.5 85-89 87 2 15 0.10 0.75

79.5-74.5 80-74 82 3 13 0.15 0.65

74.5-79.5 75-79 77 1 10 0.05 0.50

69.5-74.5 70-74 72 3 9 0.15 0.45

64.5-69.5 65-69 67 2 6 0.10 0.30

59.5-64.5 60-64 62 0 4 0.00 0.20

54.5-59.5 55-59 57 3 4 0.15 0.20

49.5-54.5 50-54 52 1 1 0.05 0.05

Total 20 1.00*

*Note that the final cumulative relative frequency score should equal thetotal relative frequency score.


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IX. Cumulative Percent

Cumulative Percent is simply the Cumulative Relative Frequencymultiplied by 100. The cumulative percent is shown in the table below:

RealLimit

s

Apparent Limits

Midpoints

Frequency

Cumulative

Frequency

RelativeFrequenc

y

Cumulative RelativeFrequency

Cumulative Percent

94.5-99.5

95-99 97 3 20 0.15 1.00 100*

89.5-94.5

90-94 92 2 17 0.10 0.85 85

84.5-89.5

85-89 87 2 15 0.10 0.75 75

79.5-74.5

80-74 82 3 13 0.15 0.65 65

74.5-79.5

75-79 77 1 10 0.05 0.50 50

69.5-74.5

70-74 72 3 9 0.15 0.45 45

64.5-69.5

65-69 67 2 6 0.10 0.30 30

59.5-64.5

60-64 62 0 4 0.00 0.20 20

54.5-59.5 55-59 57 3 4 0.15 0.20 20

49.5-54.5

50-54 52 1 1 0.05 0.05 5

Total 20

*Note that the final cumulative percent score should equal 100%.

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Cummulative Frequency Distribution