Frequency Distribution

Dhon G. Dungca

Frequency Distribution

• The organization or arrangement of raw data in table form showing its classes and frequencies.

• Raw or Ungrouped Data– Data collected in original form.

• Frequency– The number of times a certain

value or class of values occurs.

Example of a Frequency Distribution

Monthly Salary

Number of employees

5100-55995600-60996100-65996600-70997100-75997600-8099

1015221396

n = 75

Components of a Frequency Distribution

• Class Limits– The end numbers of a class. It is the

highest and the lowest values that can go into each class.

• Class Boundaries– Are the “true” class limits defined by

lower and upper boundaries. The lower boundaries can be determined by getting the average of the upper limit of a class and the lower limit of the next class.

• Class Interval (I)– It is simply the length of the class. It is the

difference or distance of the upper and lower class limits.

Components of a Frequency Distribution

• Class Size (C)– The difference between the upper and lower

boundaries of any class. It is not the difference between the upper and lower limits of the same class.

• Class Mark (x)– Also known as the class midpoint. It is found by

adding the upper and lower limits and dividing by two. It can also be found by adding the upper and lower boundaries and dividing by two.

• Cumulative Frequency (F)– The number of values less than the upper class

boundary for the current class. This is a running total of the frequencies.

• Relative Frequency– The frequency divided by the total frequency. This

gives the percent of values falling in that class.

Constructing a Frequency Distribution

• Array the given raw data in ascending order.• Find the largest and smallest values. Compute

the Range = Maximum - Minimum• Compute for the tentative number of classes (K).

The ideal number of classes is between 5 and 15. Or you may use the Sturges Method:

K = 1 + 3.33 log n

Where: K = tentative number of classesn = total number of observationslog = common logarithm (base

10)

Constructing a Frequency Distribution

• Compute for the Class Interval (I) by dividing the range by the tentative number of classes (K). Use class interval rounded to the nearest whole number.

• Sort the arrayed data into appropriate classes using convenient and easy to read class limits. Start the first class with a lower limit either equal to or a little bit less than the lowest observed value.

• Set up the class boundaries if necessary.• Determine the class mark or midpoint.• Count or tally the number of observations into

the appropriate class intervals.• If necessary, find the relative frequencies and/or

relative cumulative frequencies.

Example 1

• The following are the scores obtained by 35 students in a quiz in statistics:

• Construct a frequency distribution showing the class limits, class boundaries, the class mark, the relative frequency and the cumulative frequency if K = 7.

100 42 82 65 72 80 67

95 64 93 90 78 72 54

35 55 45 83 47 88 73

70 69 57 98 65 66 55

61 73 77 85 60 51 80

Graphical Representation of the Frequency

Distribution

• Histogram– A graph which displays the data by using

vertical bars of various heights to represent frequencies. The horizontal axis are the class boundaries.

• Frequency Polygon– A closed broken line curved constructed by

plotting the class marks on the horizontal or x-axis against the class frequencies which are plotted on the vertical or y-axis.

• Ogive (pronounced o-jive)– Is the graph of a cumulative frequency

distribution. It is constructed by plotting the class boundaries on the horizontal against the cumulative frequencies plotted on the vertical axis.

Graphical Representation of the Frequency

Distribution

0

2

4

6

8

10

Histogram

Graphical Representation of the Frequency

Distribution Frequency Polygon

0

2

4

6

8

10

12

29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.5 110

Class Marks

Freq

uenc

y (f)

Graphical Representation of the Frequency

Distribution

Ogive

0

5

10

15

20

25

30

35

40

Example 2

• The following data taken from the records of the personnel department of a bank in Makati are the scores obtained in an aptitude test given to 50 applicants:

• Prepare a frequency distribution table showing the following:

– The class limits, class boundaries and class marks.

– The relative frequency and the cumulative frequencies.

– Construct the frequency polygon for this distribution.

75 45 97 75 70 67 56 53 55 58

48 51 76 32 87 58 71 35 82 55

65 55 62 50 42 89 65 45 73 78

67 65 55 35 64 70 85 63 70 89

75 59 49 54 72 73 85 75 45 93