- This document discusses descriptive statistics for ungrouped data, including how to organize raw data into a frequency distribution and present it graphically. It also defines measures of central tendency like mean, median and mode, as well as measures of dispersion like range, variance and standard deviation. Several examples and exercises are provided to illustrate calculating these common statistical measures for ungrouped data sets.

1. • If you don’t drive your business, you will be driven out of the business.
• (B. C. Forbes)
Preparedby:WAN NURULHUDA
FACULTY OF BUSINESS, FINANCE &
IT & SCHOOL OF HOSPITALITY
MANAGEMENT
MAHSAUniversity @2020
LESSON 4:
DESCRIPTIVE STATISTICS:
UNGROUPED DATA

2. LEARNING OBJECTIVES
The study of this chapter should enable you to:
• Organize raw data into a frequency distribution
• Present a frequency distribution into graphic forms
• Describe and calculate different measures of central
tendency
• Define and calculate different measures of dispersion and
skewness

3. • Descriptive statistics aims to summarize quantitative data without using the probabilistic
formulation and not to draw conclusions about the population as in inferential statistics.
• Although a data analysis uses inferential statistics to deduce the main conclusions, the
features of descriptive statistics are usually highlighted at the same time.
• The computer has become an important tool in the presentation and analysis of data.
Among the most popular and widely used statistical packages are SAS, SPSS, Stat
graphics and Minitab.
• The collected sample are referred to as raw data. This chapter discusses to how to
organize such raw data into a frequency distribution and then present it using graphic
forms.
INTRODUCTION

4. • A frequency distribution is a tabulation of values that contains
one or more variables in a sample. It summarizes the distribution
of values in the sample where each entry represents the
frequency or count of the occurrences of values within a
particular interval.
• It is much simpler to manage and operate the frequency tabulated
data than the raw data. With frequency tables, there are simple
formulas to calculate the important statistics.
• A frequency distribution for qualitative data lists all categories
and the number of elements that belong to each of the categories.
FREQUENCY DISTRIBUTION

5. • Ungrouped data which is also known as raw data is data that has not been placed in any group
or category after collection.
• Data is categorized in numbers or characteristics therefore, the data which has not been put in
any of the categories is ungrouped.
• For example, when conducting census and you want to analyze how many women above the age
of 45 are in a particular area, you first need to know how many people reside in that area.
Some of the advantages of ungrouped data are as follows;
•Most people can easily interpret it.
•When the sample size is small, it is easy to calculate the mean, mode and median.
•It does not require technical expertise to analyze it.
WHAT IS UNGROUPED
DATA?

6. Differences between Grouped Data and Ungrouped Data
•Classification of Grouped Data vs. Ungrouped Data
Grouped data is data that has been organized in classes after its analysis. Examples include how many bags of maize collected
during the rainy season were bad. On the other hand, ungrouped data is data which does not fall in any group. It is still raw
data.
•Preference of Grouped Data vs. Ungrouped Data
When collecting data, ungrouped data is preferred because the information is still in its original form. It has not been tampered
with by classification or subdivision. However, when analyzing it and drawing graphs, grouped data is preferred because it is
simple to interpret.
•Accuracy of Grouped Data vs. Ungrouped Data
When calculating the means of grouped and ungrouped data, there will be a variation. The mean of grouped data is preferred
because it is more accurate as compared to the mean of ungrouped data. The mean of ungrouped data may lead to wrong
manipulation of the median therefore it is considered inefficient in most cases.
•Representation of Grouped Data vs. Ungrouped Data
Frequency tables are used to show the information of grouped data whereas in the case of ungrouped data, the information
appears like a big list of of numbers. This is due to the fact that the information is still raw.

8. UNGROUPED DATA
• MEAN:
➢A set of data has only one mean (unique).
➢ The mean is a useful measure for comparing two or more populations.
• MEDIAN:
➢The median for ungrouped data is the midpoint of values after they have
been arranged from the
➢smallest to the largest, or the largest to the smallest
Median : * Middle value, if n is odd;
* Mean of the two middle values, if n is even.
MODE:
➢The mode of a set of measurements is the value that occurs most
frequently.

9. Mean
 x = sum of all observation
n = total number
•Mean is the sum of all data divided by the
total number.
•Formula,
9
ҧ
𝑥=
σ 𝑥
𝑛

10. 1) Find the mean of the following data,
1, 4, 8, 15, 14, 20, 23.
2) Find the mean of the following data
122 145 136 122 111 187 118
136 121 110 154 125 166 126
EXERCISE

11. MEDIAN
•Median is the middle value of a set of
data.
•But, when data are arranged in order.
•When there is an extreme value in the
data, the median is more suitable to be
used for representing the data.
1

12. EXAMPLE
•Find the median of the following sets of data,
3, 4, 7, 3, 10, 12, 5
rearrange : 3, 3, 4, 5, 7, 10, 12
= 3, 3, 4, 5, 7, 10, 12
= 5

13. Exercise
3) Find the median of the following sets of data,
9, 7, 2, 4, 15, 10, 8
4) Find the median of the following sets of data,
91, 70, 2, 40, 15, 100, 8, 130
1
3

14. MODE
•Mode is the data value that occurs with the highest
frequency in a set of data.
•The mode is an important representation
for categorical data.
•Example:
1
4

15. 1
5
• Find the mode in the following data, 2, 7, 2, 4, 15, 4, 4, 13
• Find the mode in the following data, 12, 7, 32, 14, 15, 44, 14, 113,
2
• Find the mode in the following data, 2, 7, 2, 4, 15, 4, 4, 13, 2
EXAMPLE

16. EXERCISE (CONT.)
5) Find the mode in the following data, 112, 37, 32, 14, 15, 44, 4,
113
6) Find the mode in the following data, 22, 77, 15, 22, 44, 15, 44, 77
16

17. • QUARTILE:
➢Descriptive measures that split the ordered data into four quarters.
➢INTERQUARTILE:
➢QUARTILE DEVIATION:
• RANGE:
➢Largest value- smallest value
4
1
1
+
=
n
Q
( )
4
1
3
3
+
=
n
Q
𝑄3 − 𝑄1
𝑄3 − 𝑄1
2

18. EXERCISE
a) Find the quartile, interquartile and quartile deviation in the
following data, 112, 37, 32, 14, 15, 44, 4, 113
b) Find the quartile, interquartile and quartile deviation in the
following data, 22, 77, 15, 22, 44, 15, 44, 77
16

19. • VARIANCE:
• STANDARD DEVIATION:
𝑠2 =
1
𝑛 − 1
෍ 𝑥2 −
σ 𝑥
𝑛
2
𝑠 = 𝑣𝑎𝑟

20. 151 120 122
112 100 105
• During a 6 days period, Saiful’s daily incomes were as follows
i. 1st and 3rd quartile
ii. interquartile range
iii. Quartile deviation
iv. Range
v. Variance
vi. Standard deviation

21. EXERCISE 1:
• A company has six persons for Family day. The number of
player in six persons are : 10, 12, 15, 12 , and 13
respectively.
Calculate:
i. Mean
ii. Median
iii. Mode
iv. 1st and 3rd quartile
v. interquartile range
vi. Quartile deviation
vii.Range
viii.Variance
ix. Standard deviation

22. EXERCISE 2:
• Ms Rachel collects the data on the ages of mathematics
teacher in Monte Carlo School, and her study yields the
following data:
38 35 28 36 35 33 40
Calculate:
i. Mean
ii. Median
iii. Mode
iv. Variance
v. Standard deviation

23. EXERCISE 3:
• For the sample below:
Calculate:
i. Range
ii. Interquartile
iii. Quartile deviation
15 26 38 72 66 91
87 93 100 52 44 38

24. Preparedby:WAN NURULHUDA
FACULTY OF BUSINESS, FINANCE &
IT & SCHOOL OF HOSPITALITY
MANAGEMENT
MAHSAUniversity @2020