UNGROUPED DATA
Introduction
Fields of Statistics
1. Applied Statistics is concerned with the
procedures and techniques used in collection,
presentation, organization, analysis and interpretation
of data.
2. Theoretical statistics is concerned with the
development of the mathematical foundations of the
methods used in applied statistics.
Areas of interest in Applied Statistics
1. Descriptive statistics includes all the techniques
used in organizing, summarizing and presenting the
data on hand.
2. Inferential statistics includes all the techniques used
in analysing the sample data that will lead to
generalizations about a population from which the
sample came from.
Descriptive Statistics
Measures of Central Tendency are descriptive measures
that are used to describe the center of a set of data,
arranged numerically.
1. The arithmetic mean is the most commonly used
measure of central tendency. When we speak of average,
we always refer to the mean.
2. The median is the value that divides the array into two
equal parts. It is the midpoint of the data array.
3. The mode is the observed value that occurs with the
greatest frequency in a data set.
Measures of Central Tendency for Ungrouped Data
Finding the mean
To find the mean of ungrouped data use :
𝚺𝒙
x=
𝑵
Where :
x = mean
𝚺𝒙 = sum of the given observations
N = number of observations
Measures of Central Tendency for Ungrouped Data
Examples
Six friends in a Mathematics class receives test grades of 92,
84, 65, 76, 88, and 90. Find the mean of these scores.
𝚺𝒙
x=
𝑵
𝟗𝟐 + 𝟖𝟒 +𝟔𝟓 + 𝟕𝟔 + 𝟖𝟖 + 𝟗𝟎
=
𝟔
495
=
6
x = 82. 5
Measures of Central Tendency for Ungrouped Data
Examples
The ages of five contestants in a Statistics Quiz Bee are the
following: 18, 17, 18, 19, and 18. Find their average age.
𝚺𝒙
x=
𝑵
𝟏𝟖 +𝟏𝟕 +𝟏𝟖 +𝟏𝟗 +𝟏𝟖
=
𝟓
90
=
5
x = 18
Measures of Central Tendency for Ungrouped Data
Finding the median
Before finding the median, the data must be arranged in order, from least to greatest
value. The median will be a specific value or will fall between two values.
 If the number of data is odd, the median is the middle value
 If the number of data is even, add the two middle values then divide the sum by 2.
Therefore, the median is
𝒙𝟑 + 𝒙𝟒
=
𝟐
Measures of Central Tendency for Ungrouped Data
Examples
Seven mothers were selected and given a blood pressure check.
Their systolic blood pressure were recorded as follows: 135 121
119 116 130 121 131. Find the median.
Solution: arrange the data in order
116, 119, 121, 121, 130, 131, 135
There are seven observations in the data, and 7 is an odd number
so the median is the middle value which is the fourth value.
Therefore, the median ,
= 121
Measures of Central Tendency for Ungrouped Data
Examples
Find the median of the following:
5, 5, 2, 7, 9, 10, 7, 8, 6 , 14, 20, 25
Solution: arrange the data in order:
2, 5, 5, 6, 7, 7, 8, 9, 10, 14, 20, 25
There are twelve observations in the data, and 12 is an even
number so the median are the two middle values which are the 6th
and the 7th value. Therefore, the median ,
𝒙𝟔 + 𝒙𝟕
𝟕 + 𝟖 𝟏𝟓
=
=
=
=
𝟕.
𝟓
𝟐
𝟐
𝟐
Measures of Central Tendency for Ungrouped Data
Finding the mode
The mode is the value that occurs most often in a data set. If
there is only one mode the set is unimodal; if there are 2 modes,
then it is bimodal. More than 2 modes shows a data.
Example:
Find the mode of the given data: 15, 28, 25, 48, 22, 43, 39, 44, 43,
49, 22, 33, 27, 25, 22, and 30.
The number that appeared most is 22. Therefore the mode is 22.
The data is unimodal.
Measures of Central Tendency for Ungrouped Data
Example:
Find the mode of the given data: 121, 110, 120, 119, 121, 118,
115, 107, 115,
The numbers that appeared most are 115 and 121 . Thus, the
data has two modes, and the data is said to be bimodal.
Measures of Central Tendency for Ungrouped Data
The Weighted Mean
Weighted mean - is a type of mean that is calculated by multiplying the
weight (or probability) associated with a particular event or outcome with its
associated quantitative outcome and then summing all the products
together.
To find the mean of ungrouped data use :
𝚺 (𝒙∙𝒘)
x=
𝚺𝒘
Where :
x = weighted mean
𝚺 (𝒙 ∙ 𝒘)= sum of the products formed by multiplying each number by its assigned
weight.
𝚺𝒘 = the sum of all the weights
Measures of Central Tendency for Ungrouped Data
What is the weighted mean of the ff:
Subject
Grade (x)
Units (w)
(𝒙 ∙ 𝒘)
Mathematics
3.00
3
9.0
English
2.00
3
6.0
P.E.
1.25
2
2.5
𝜮𝒘=8
𝜮 (𝒙 ∙ 𝒘) = 17. 5
𝚺 𝒙∙𝒘
x=
𝚺𝒘
𝟏𝟕.𝟓
=
= 𝟐. 𝟏𝟖𝟕𝟓 ≈ 𝟐. 𝟏𝟗
𝟖