Measures-of-Variation-part-1 (1).pptx

INTRO TO MATHEMATICAL STATISTICS

MEASURES OF VARIATION
• also known as “Measures of Dispersion”
• Suppose you ask a group of college
students to rate the quality of food at the
school canteen and you find out that the
average rating is 3.5 using the following
scale: 5 (excellent); 4 (Very Satisfactory); 3
(Satisfactory); 2 (fair); and 1 (poor).
• How close are the ratings given by the
students? Do their ratings cluster around
the middle point of 3, or are their ratings
spread or dispersed, with some students
giving ratings of 1 and the rest giving
ratings of 5?

The extent of the spread, or the dispersion of
the data is described by a group of measures
called measures of dispersion, also called
measures of variability.
The measures to be considered are the
range, mean deviation, variance and
standard deviation.

1. The Range of a Set of Data
• The range of a set of data is the difference
between the highest and lowest values in the
set. To find the range, first order the data from
least to greatest. Then subtract the smallest
value from the largest value in the set.

1. The Range of a Set of Data
Examples:
• 1. John Carlo took 7 math tests in one
marking period. What is the range of his test
scores?
89, 73, 84, 91, 87, 77, 94
Solution: Ordering the test scores from least to
greatest, we get:
73, 77, 84, 87, 89, 91, 94
Highest – lowest = 94 – 73 = 21
჻ The range of these test scores is 21 points.

2. Average or Mean Deviation
• is the average of the absolute values of the
deviation scores.; that is, mean deviation is
the average distance between the mean and
the data points. Closely related to the
measure of mean deviation is the measure of
variance.
• In Mathematics, the term “absolute”
represented by the sign “‫׀‬ ‫“׀‬ simply means
taking the value of a number without regard to
positive or negative sign.

For Ungrouped Data:
Average Deviation (AD) =
𝒙−𝒙
𝐧
Example:
Consider a set of values which
consists of 20, 25, 35, 40, 45. Solving for the
mean,
𝟐𝟎+𝟐𝟓+𝟑𝟓+𝟒𝟎+𝟒𝟓
𝟓
= 33.

For Ungrouped Data:
𝒙−𝒙
𝐧
Solving for AD:
AD =
𝟐𝟎−𝟑𝟑 + 𝟐𝟓−𝟑𝟑 + 𝟑𝟓−𝟑𝟑 + 𝟒𝟎−𝟑𝟑 +|𝟒𝟓−𝟑𝟑
𝟓
=
−𝟏𝟑 + −𝟖 + 𝟐 + 𝟕 +|𝟏𝟐
𝟓
=
𝟏𝟑+𝟖+𝟐+𝟕+𝟏𝟐
𝟓
=
𝟒𝟐
𝟓
= 8.4
Thus, on the average, each value is 8.4 units
from the mean.

For Ungrouped Data:
𝒙−𝒙
𝐧
Example 2:
A set of observations consists of 22, 60, 75,
85, 98. Find the average deviation.

Example 2:
A set of observations consists of 22, 60, 75,
85, 98. Find the average deviation.
𝒙 =
𝟐𝟐+𝟔𝟎+𝟕𝟓+𝟖𝟓+𝟗𝟖
𝟓
= 68
So, AD =
𝟐𝟐−𝟔𝟖 + 𝟔𝟎−𝟔𝟖 + 𝟕𝟓−𝟔𝟖 + 𝟖𝟓−𝟔𝟖 +|𝟗𝟖−𝟔𝟖
𝟓
=
−𝟒𝟔 + −𝟖 + 𝟕 + 𝟏𝟕 +|𝟑𝟎
𝟓
=
𝟒𝟔+𝟖+𝟕+𝟏𝟕+𝟑𝟎
𝟓
=
𝟏𝟎𝟖
𝟓
= 21.6
Thus, on the average, each value is 21.6
units from the mean.

3. Variance
It also indicates a relationship between the mean
of a distribution and the data points; it is
determined by averaging the sum of the squared
deviations. Squaring the differences instead of
taking the absolute values allows for greater
flexibility in calculating further algebraic
manipulations of the data.

3. Variance
• For Population Variance:
𝞼2 =
(𝑥−𝞵)2
𝑁
Where:
𝞼 = (sigma squared) , population variance
(𝑥 − 𝞵)2 = square of the differences
(𝑥 − 𝞵)2
= sum of squared differences
N = the number of scores

3. Variance
Example:
Worksheet for Calculating the Variance for 7
scores
X Solving for 𝞵;
5
3 𝞵 =
28
7
4 𝞵 = 4
4
3
4
5
28

3. Variance
Example:
scores ; 𝞵 = 4
X X - μ (X - μ)2
5 1 1
3 -1 1
4 0 0
4 0 0
3 -1 1
4 0 0
5 1 1
28 4

3. Variance
Example:
scores ; 𝞵 = 4
𝞼2=
(𝑥−𝞵)2
𝑁
=
4
7
= 0.57

3. Variance
Example:
Considering the data as a sample:
scores ; 𝞵 = 4
𝑠2=
(𝑥−𝑥 )2
𝑛−1
=
4
7−1
=
4
6
= 0.67
For our problem then, the sample variance is
0.667 (slightly larger than the population
variance).

3. Variance
Practice:
Find the variance of the given data ( considering
the data as sample):
15, 18, 22 , 22, 25, 18, 30, 35, 40, 45

3. Variance
Calculating the variance by the Raw Score
Method
• When we actually calculate the variance for a
set of scores, we generally use the raw score
method. This is also the method used by
computers or calculators to find the variance.
• The formula for the variance by the raw score
method is mathematically equivalent to the
deviation score method.
• The raw score formula for the variance is:
𝞼2=
𝑥2−
( 𝑥)2
𝑁
𝑁

3. Variance
Calculating the variance by the Raw Score
Method
The raw score formula for the variance is:
𝞼2=
𝑥2−
( 𝑥)2
𝑁
𝑁
We can see that from the formula to find the
population variance we sum up the squared
individual scores and subtract from the sum of
the scores quantity squared divided by the
number of scores.

3. Variance
Example:
Scores of 7 Students in an
Exam (x)
5
3
4
4
3
4
5

3. Variance
Example: Scores of 7
Students in an
Exam (x)
𝒙𝟐
5 25
3 9
4 16
4 16
3 9
4 16
5 25
⅀ 28 116

3. Variance
Example:
Solution:
𝞼2=
𝑥2−
( 𝑥)2
𝑁
𝑁
𝞼2=
116−
(28)2
7
7
𝞼2=
116−
784
7
7
𝞼2=
116−112
7
𝞼2=
4
7
= 0.57

3. Variance
Example: if we were to consider the given
scores as a sample, we would have-
Solution:
𝑠2=
𝑥2−
( 𝑥)2
𝑁
𝑛−1
𝑠2=
116−
(28)2
7
7−1
𝑠2
=
116−
784
7
6
𝑠2=
116−112
6
𝑠2=
4
6
= 0.67

3. Variance
Practice: Using the raw score method
Find the variance of the given data ( considering
the data as population and as a sample):
15, 18, 22 , 22, 25, 18, 30, 35, 40, 45

Measures-of-Variation-part-1 (1).pptx

Recommended

Recommended

More Related Content

Similar to Measures-of-Variation-part-1 (1).pptx

Similar to Measures-of-Variation-part-1 (1).pptx (20)

Recently uploaded

Recently uploaded (20)

Measures-of-Variation-part-1 (1).pptx