Unit 6: Topic 2: Properties of Variance & Standard Deviation
Competencies
·
explain
the properties of variance and standard deviation
·
use
the properties of variance and standard deviation to solve related problems
Suggested ways of teaching this topic: Brainstorming
and Guided Discovery
Starter
Activities
The
teacher might start with the following brainstorming questions to revise the
previous lesson.
·
What is Standard
deviation?
·
What do we mean
by deviation?
· "What is the Variance?"
·
How can we find
Variance of a given data? The Standard deviation?
Expected
Responses
The Standard Deviation is a
measure of how spreads out the numbers are.
Its symbol is σ (the greek letter sigma)
The formula is easy: it is the square root of the Variance.
Deviation just means how far from the normal
The Variance is defined as: The average of the squared differences from the Mean.
To calculate the variance, we follow the following steps:
- Work out
the Mean (the simple average of the numbers)
- Then for
each number: subtract the Mean and square the result (the squared
difference).
- Then work
out the average of those squared differences. (Why Square?)
Example: Given the
population function 2, 4, 1, 5, find the variance and standard deviation. Then
add 3 to each data item, and find the variance and standard deviation of the
resulting population function compare the variance values and standard
deviation values.
Solution: Data items: 2, 4, 1, 5
= 
= 
= 3
d2 =
= 
=
=
2.5
Old data items:
2, 4, 1, 5
New data items:
5, 7, 4, 8
New 
=
= 
= 6
Newd2 =
= = = 2. 5
Thus, the new mean is 3 more than the
old mean. Both variance and standard deviations are the same.
·
At
this stage the teacher should ask students: “What can we conclude from this
example?” The answer is the following property.
Property 1
If a constant c is added to each value
of a population function, then the new variance is the same as that of the old
variance. The new standard deviation is also the same as that of the old standard
deviation.
·
The
teacher could ask students to prove this property.
Proof of
Property 1
Consider the data items:
x1, x2,
x3, . . . xn having mean.
Add c to each data item: x1 + c, x2 + c, x3
+c, . . . xn + c and the new mean
= + c
New
d2
= (x1 - )2 + (x2
- )2 +
. . . + (xn
- )2
= old d2
Note: If the values are equal, the square root
of the variance will be equal. Therefore, the standard deviations are also
equal.
Example: Given the population function, 2, 1, 4,
5, find the mean, variance and standard deviation. Then multiply each data item
by 3 and find the new mean, variance and standard deviation. Compare the old
and new mean, variance and standard deviation:
Solution:
Old
data items: 2, 1, 4, 5
= 
= = 3
d2
=
= 
= 
= 2.5
d = 
New
data items: 6, 3, 12, 15
New
mean = = 
= 
= 9
New variance = 
=
= 
= 
= 22.5
d = 
The
new mean = 
= 9 = 3 ´ 3 = ´ 3
The
new variance = 22.5 = 9 ´ 2.5 = 32´ 2.5
=
32´ the old variance.
The
new standard deviation = = 
= 
= 
= 
the old st. deviation
·
At
this stage the teacher should ask students: “What can we conclude from this
example?” The answer is the following property.
Property 2
If each data item of a population
function is multiplied by a constant c, the new variance is c2 times
the old variance. The new standard deviation is |c| times the old standard
deviation.
·
The
teacher could ask students to proof this property for the general case
Proof of
Property 2
Consider
the population function x1, x2, x3, . . . xn whose mean is and variance d2 .
New data items:
cx1, cx2,
cx3, . . . cxn having new mean
= c
.
New
variance =
= = 
= c2´ the old
variance
New standard deviation = 
Concluding Activities
The teacher
shall make sure that the following points are well taken and summarized by the
students.
§ If all the
values of a population are increased by a constant c then, the mean is also
increased by c while the standard deviation remains unchanged.
§ If all the
values of a population are multiplied by a constant c then,
i)
The
new mean is c the old mean
ii)
The
new standard deviation is |c| the old standard deviation
Practice Exercises
1. What is the
standard deviation for the numbers: 75, 83, 96, 100, 121 and 125?
2. Find the
variance and standard deviation of the given population function:- 3, 5, -1, 3.
Then subtract 2 from each data item, and find the variance and standard
deviation of the new data items. Compare
the old and new variance values and standard deviation. What can you conclude?
3. Given the
population function, 2,1,3,2, find the mean, variance and standard deviation.
Then multiply each data item by – 2 and find the new mean, variance and
standard deviation. Compare the old and new mean, variance and standard
deviation.
4.
Use
the given frequency distribution table to find the mean, variance and standard
deviation of the population function.
|
Value |
-2 |
0 |
2 |
3 |
4 |
|
Frequency |
1 |
2 |
2 |
2 |
3 |
5. What is the
standard deviation of the first 10 natural numbers (1 to 10)?
6. The standard
deviation of the numbers 3, 8, 12, 17 and 25 is 7.56 correct to 2 decimal
places. What will be the new standard deviation if every value is:
a) Increased by 4?
b) Decreased by 2?
c) Multiplied by 4?
d) Halved?