Measures of Variability: Variance and Standard Deviation
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Measures of
Variability
Variance
and
Standard Deviation
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from parents
Measures of
Variability
Another measure of the variability in a
data set uses the
deviations
from the
mean (
x
–
x
).
Remember the sample of 6 fish that we caught from
the lake . . .
They were the following lengths:
3”, 4”, 5”, 6”, 8”, 10”
The mean length was 6 inches. Recall that we
calculated the deviations from the mean.
What was the
sum
of these deviations?
Can we find an average deviation
?
What can we do to the deviations so that
we could find an average
?
The estimated average of the deviations
squared
is called the
variance
.
Standard Deviation
-
is the square root of the variance.
-
is the average distance from the
center(mean).
Notations
Degree of
freedom
When calculating sample variance, we use degrees of freedom (
n
– 1)
in the denominator instead of
n
because this tends to produce
better estimates
.
Degrees of freedom will be revisited again in Chapter 8.
What is the sum
of the deviations
squared?
Remember the
sample
of 6 fish that we caught from the lake . . .
Find the variance of the length of fish.
Divide this by 5.
First square the
deviations
34
s
2
= 6.8
A
typical deviation from the mean is the
standard deviation.
s
2
= 6.8 inches
2
so
s = 2.608
inches
The fish in our sample
deviate
from the
mean
of
6 by an average of
2.608
inches.
The most commonly used measures of
center
and
variability
are the
mean
and
standard deviation
, respectively.
Choosing Measures of Center and Spread
-
M
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a
n
a
n
d
S
t
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d
a
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d
D
e
v
i
a
t
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n
-
M
e
d
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a
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d
I
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e
r
q
u
a
r
t
i
l
e
R
a
n
g
e
•
The
median
and
IQR
are usually better than
the mean and standard deviation for
describing a
skewed
distribution or a
distribution with outliers.
•
Use
mean
and
standard deviation
only for
reasonably
symmetric
distributions that don’t
have outliers.
Rule of Thumb
The
range
is
4
times as much as the
standard deviation
.
How to use
GDC
?
TI-nspire
1.
Go to
Lists & spreadsheet
2. Give a
name
for the column
3. Enter the data under this column
4. Press
Menu
5. Choose
(4) Statistics
6. Select
Stat Calculation
7. Select
one-variable statistics
8. Enter
9. Num of Lists 1 (press ok)
10. x1 List ……. The name of the column where you
enter the data
11. Leave others as it is
12. Press ok
Example:
Pair-work
Homework
Page-71
98
100
103
Wednesday, 18
th
Test
on
Chapter-3
This lesson covers the concepts of variance and standard deviation as measures of variability in a data set. It explains how deviations from the mean are used to calculate variance, and how standard deviation, as the square root of variance, measures the average distance from the mean. Degree of freedom in sample variance calculation is also discussed. Examples with a sample of fish lengths are used to illustrate these concepts.
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Presentation Transcript
Calculator required for this lesson!
Measures of Variability Variance and Standard Deviation
Awesome text messages from parents
Measures of Variability Another measure of the variability in a data set uses the deviations from the mean (x x).
Remember the sample of 6 fish that we caught from the lake . . . They were the following lengths: 3 , 4 , 5 , 6 , 8 , 10 The mean length was 6 inches. Recall that we calculated the deviations from the mean. What was the sum of these deviations? Can we find an average deviation? What can we do to the deviations so that we could find an average?
The estimated average of the deviations squared is called the variance. N ( ) 2 x-m s2=
Standard Deviation - is the square root of the variance. ( ) 2 x-m N s = - is the average distance from the center(mean).
Notations s for populationstandard deviation s for sample standarddeviation
2 ( ) x x = 2 s n 1 Degree of freedom When calculating sample variance, we use degrees of freedom (n 1) in the denominator instead of n because this tends to produce better estimates. Degrees of freedom will be revisited again in Chapter 8.
Remember the sample of 6 fish that we caught from the lake . . . Find the variance of the length of fish. First square the deviations x 3 4 5 6 8 10 Sum (x - x) -3 -2 -1 0 2 4 0 (x - x)2 9 4 1 What is the sum of the deviations squared? Divide this by 5. 0 4 16 34 s2= 6.8
A typical deviation from the mean is the standard deviation. s2= 6.8 inches2so s = 2.608 inches The fish in our sample deviate from the mean of 6 by an average of 2.608 inches.
The most commonly used measures of center and variability are the mean and standard deviation, respectively.
Choosing Measures of Center and Spread - Mean and Standard Deviation - Median and Interquartile Range
The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don t have outliers.
Rule of Thumb The range is 4 times as much as the standard deviation.
TI-nspire 1.Go to Lists & spreadsheet 2. Give a name for the column 3. Enter the data under this column 4. Press Menu 5. Choose (4) Statistics 6. Select Stat Calculation 7. Select one-variable statistics 8. Enter 9. Num of Lists 1 (press ok) 10. x1 List . The name of the column where you enter the data 11. Leave others as it is 12. Press ok
Example: This is the key phrase to solve!
Homework Page-71 98 100 103
Wednesday, 18th Test on Chapter-3
