Example
:
find the sample mean and the range
1.
3, 3, 3, 3, 3.
2.
1, 2, 3, 4, 5.
3.
1, 1, 3, 5, 7.

Variance
Population variance
Sample variance
Standard deviation
for the population
for the sample
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Note:
Population variance :
Let
x , x ,….x. be a population of size N, with mean μ
, then the population
variance is given by
Example
:
evaluate
1, 2, 3, 4, 5.
Example
:
evaluate
1, 1, 3, 5, 7.
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Sample variance
Let
x , x ,….x. be a population of size n, with mean
x , then the sample
variance is given by
Example
: evaluate
10, 12, 11, 13, 15

Another formula for
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Variance for frequency table
Example
: find
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Age
Frequency
18
2
19
6
20
9
21
4
22
3
23
1
Total
Example
: find the variance and Standard deviation
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grades
Frequenc
y
1

5
6
6

10
9
11

15
12
16

20
3
Total
Other descriptive measures
Coefficient of variation is used to compare the variability of two data
sets or more.
Coefficient of variation
C.V=
Example:
which sample has more variability?
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Sample 1
sample 2
age
10 years
25 years
Mean weight
35 kg
75 kg
variance
200 kg
200kg
K
th
percentiles
P : is the value where k% of data less than it
Example:
Given the following data
46 ,57,59 ,61 ,62 ,70 ,73 ,77,80 ,81 ,82 ,82 ,84 ,86 ,90 ,91 ,93, 95, 96.
a.
find the 20
th
percentiles
b.
find the
P
c.
Find
P
d.
Find
P
e.
Find
P
f.
Find
P
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To find P
1.
Order the data from
smallest to largest .
2.
Determine the
position of the
percentile
L=
Percentiles and Quartiles
Quartiles divide the data into 4 equal parts.
o
P
:
First quartile(Q
)
means that
25% of the data less than
or equal
to Q
or 75% of the data more
than or equal to Q
o
P
:
Second quartile (Q ) =
median
means that
50% of the data less
than
or equal to Q
Or 50% of the data more
than or equal to Q
o
P
:
Third quartile (Q )
means that
75% of the data less than
or equal
to Q
Or 25% of the data more
than or equal to Q
Inter quartile
range (IQR)= Q
 Q
Is used to measure the variability when the data set is significantly sewed .
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Box and whisker plot
To construct the Box
–
plot we need the following
values
max /min /Q
/Q
/Q
.
Example
:
construct the box
–
and whisker plot for the following data.
7,
3,
14,
9,
7,
8,
12
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Example
:
construct the box
–
and whisker plot for the following data.
19,
21,
47,
55,
78,
65,
56, 6 15, 24, 38.
Order the date from smallest to largest……………………………….
Min=………….
Max=…………….
P
= Q
P
= Q
P
=Q
Box Plot
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Skewed Data
Data can be “ skewed” meaning it tends
to have along tail on one side
than the other .
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The normal distribution No skew
A normal distribution is not skewed it is
perfectly symmetrical and
the mean
is
exactly at the peak
Positive skew
It is when the long tail is on the positive side of the peak, and some people say
it is “skewed to the right “
Negative skew
A Negative skew is when the long tail
is on the negative side of the peak,
people say it is “ skewed to the left”
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 Standard Deviation, Ms. Lubna Ineirat