MATHEMATICS IN THE MODERN WORLD
MODULE 4
I. TOPIC: STATISTICS (FREQUENCY DISTRIBUTION, RELATIVE FREQUENCY)
II. OBJECTIVE(S):
1. Make a frequency table for a set of data
2. Create a frequency distribution for a data set
3. Understand the relative frequency distribution table
III. INTRODUCTION:
Statistics (in the singular sense) is a scientific discipline that deals
with the methods and theories in the manipulation of numerical data. It
leads to the analysis and interpretation of the data set so one can make a
sound decision and thorough inferences.
Statistics (in the plural sense) are numerical data. Some examples are
revenues, allowed kilograms for check in luggage, stipend, tuition fee, ID
number, military ranks, etc.
IV. DISCUSSION: DATA MANAGEMENT
Data Management deals with the collection, organization and
presentation of the numerical data or (statistics) in a presentable and usable
manner.
The Frequency Distribution Table (FDT)
An FDT is a statistical table showing the frequency or number of
observations contained in each of the defined classes or categories.
Types of FDT
1. Qualitative (or Categorical) FDT – an FDT where the data are
grouped according to some qualitative characteristics; data are
grouped into non-numerical categories.
Example 1:
Frequency Distribution of the Gender of Respondents in a
Survey
Gender of Number of
respondents respondents
Male 38
Female 62
Total 100
Example 2 Distribution of Blood Types
Twenty-five army inductees were given a blood test to
determine their blood type. The data set is
A B B AB O
O O B AB B
B B O A O
A O O O AB
AB A O B A
Construct a frequency distribution for the data.
Relative
Blood Type Tally Frequency
Frequency
A
B
O
AB
A relative frequency (rf) is the percentage of items per category.
𝑓
𝑟𝑓 = × 100%
𝑁
2. Quantitative (or Numerical) FDT – the data are grouped according
to some numerical or quantitative characteristics
Example:
Frequency Distribution of the Weights of 50 pieces of Luggage
Weight (in Frequency
Kilograms)
7–9 2
10 – 12 8
13 – 15 14
16 – 18 19
19 – 21 7
Total 50
Steps in the Construction of an FDT
1. Decide on the number of class intervals k, usually between 5 and 20.
2. Determine the range R.
R = Highest Value – Lowest Value
3. Determine the class size i by the formula i = R/k. Round up the
interval size if there’s decimal.
4. Select a starting point for the lowest class limit. The first lower limit
must be less than or equal to the minimum value in the data set.
5. List all the class limits and by adding the class width to the limits of
the previous interval.
6. Find the class boundaries, also called the true limits. The boundaries
are half-way between the upper limit of one class and the lower limit
of the next class.
7. Tally the frequencies for each class.
8. Sum the frequency column and check against the total number of
observations.
9. Determine the class marks of each interval by averaging the class limits
or the class boundaries.
10. Find the relative frequency if needed.
11. Determine the cumulative frequencies, if needed.
a. less than cumulative frequency (<cf) – total number of
observations less than the upper boundary of a class interval
b. greater than cumulative frequency (>cf) – total number of
observations greater than the lower boundary of a class interval
Example 3
Construct the FDT for the following scores of 30 students in their first
quiz in statistics using five classes.
45 38 49 22 22 34
37 44 27 28 38 35
24 49 37 33 26 43
37 29 40 39 37 30
26 41 20 36 46 35
Class Class Class Relative
Tally Frequency
Frequency <cf >cf
Intervals Boundaries Marks
Graphical Representation of the Frequency Distribution Table
1. Frequency Histogram – a bar graph that displays the classes on the
horizontal axis and the frequencies of the classes on the vertical axis;
the vertical lines of the bars are erected at the class boundaries and
the height of the bars correspond to the class frequency.
2. Frequency Polygon – a line chart that is constructed by plotting the
frequencies at the class marks and connecting the plotted points by
means of straight lines; the polygon is closed by considering an
additional class at each end and the ends of the lines are brought down
to the horizontal axis at the midpoints of the additional classes.
Frequency Histogram
Frequency Polygon
When the range of the data values is relatively small, a frequency
distribution can be constructed using single data values for each class.
This type of distribution is called an ungrouped frequency
distribution.
Example The data shown here represent the number of miles per gallon
(mpg) that 30 selected four-wheel-drive sports utility vehicles obtained
in city driving. Construct a frequency distribution.
12 17 12 14 16
18 16 18 12 16
17 15 15 16 12
15 16 16 12 14
15 12 15 15 19
13 16 18 16 14
