Download 9.3: Variability

9.3: Variability
Mackenzie Coffman
Terminology
The standard deviation is a measure of how
spread out numbers are in a set of data.
The variance is the average of the squared
differences from the mean.
The mean is the average of the set of data.
Standard Deviation
Below is the formula to find standard deviation, represented by
the symbol σ, the greek letter sigma.
In other words,
the standard
deviation is the
square root of
the variance.
Variance
The variance is the standard deviation squared. For example, say the
standard deviation of a set of data is 2.75. The variance would be
represented as (2.75) 2 , equaling approximately 7.56
Quartiles
The 1st quartile (lower quartile) of a set of data is found through the following
(n+1)
equation:
4
The 3rd quartile (upper quartile) of a set of data is found through the following
equation:
3(n+1)
**In both equations, “n” represents
the number of terms in the set of
4
data.**
____
____
Interquartile Range (IQR)
The IQR is the difference between the upper quartile and lower quartile of a set of
data:
IQR= Q3 - Q1
Median
The median is the middle term in a set of data, which can be found
with the following equation:
(n+1)
___
2
x
0
1
2
3
4
5
6
freq
2
4
1
3
2
5
2
Number of terms “n”= 19
(2+4+1+3+2+5+2)
To the left is a frequency data table, showing “x,”
and the frequency of “x” occurring. From this
table, we can calculate standard deviation,
variance, mean, median, 1st and 3rd quartiles,
and the IQR.
Median:
(n+1)
__
2
19+1
__
2
10
Calculating Variance:
1. Find the mean of the data
_______________ =
_
19
x = 3.16
0+0+1+1+1+1+2+3+3+3+4+4+5+5+5+5+5+6+6
3.16
x
0
1
2
3
4
5
6
freq
2
4
1
3
2
5
2
Calculating Variance (cont.)
2. Multiply each frequency by (x2
2
_2
x)
.........
2(0-3.16) + 4(1-3.16) + 1(2-3.16)
+ 2(6-3.16)
2
2
= 74.53
3. Divide the result by the number of terms
74.53
___
=
=
Variance
3.92
19
Calculating Standard Deviation
√(3.92) = 1.98
0
1
2
3
4
5
6
Calculating lower and upper quartiles
x
freq
Q 1 = 19+1
___ = 5th term
4
2
4
1
3
2
5
2
This means we look at the 5th term, which lies within
x=1. So, Q1 = 1
Q3 =
3(19+1)
____
=
15th term
4
This means we look at the 15th term, which lies within
x=5. So, Q 3 =
IQR= Q 3- Q
IQR=5-1
IQR= 4
5
1
*Note* When calculating the
lower or upper quartiles, if the
answer is 5.25, round down to
the 5th term. If the answer is
15.75, round up to the 16th term.
[Apply the rules of rounding
when determining the term]
Box-and-Whisker Plot
One way we can organize information such as IQR and median is
through a box-and-whisker plot.
Ex: Make a box-and-whisker plot to represent the following data:
Q1= 22
Q = 57
3
IQR= 35 Median= 51
Lower Fence: Q1- 1.5 x IQR
22-1.5(35)
-30.5
Upper Fence: Q 3+ 1.5 x IQR
22
Q
1
**Not drawn to scale**
51 57
Median
Q
3
109.5
57+1.5(35)
109.5
Practice
1. Given the various heights (in inches) of first graders, find:
45, 47, 47, 47, 50, 44, 46, 45, 51, 50
[A] x
_
[B]
median
[C]
mode
____________ =47.2
10
[A] 45+47+47+47+50+44+46+45+51+50
[B]
44, 45, 45, 46, 47, 47, 47, 50, 50, 51
44, 45, 45, 46, 47, 47, 47, 50, 50, 51
47
[C]
(most re-occurring value) =
47
Age (years)
Freq
16
8
17
6
18
13
19
7
20
6
21
6
22
5
Practice (continued)
2. The following table shows the ages of participants in
the blood drive of a certain community. Find the
mean, standard deviation, and variance of the data.
Mean:
________________
16(8)+ 17(6) + 18(13) + 19(7) + 20(6) + 21(6) + 22(5)
51
= 18.69
Variance:
2
2
2
8(16-18.69) + 6(17-18.69) + 13(18-18.69)
+ 5(22-18.69) 2
.......
=
178.98
Standard Deviation:
√(178.98) = 13.38
Practice (continued)
The time (in seconds) of the top swimmers of a certain school are listed below, from
2008-2013 respectively.
x
31, 33, 28, 30, 28, 27
Find the IQR.
Q1 =
Q3
__ = 1.75
4
(6+1)
__ = 5.25
3(6+1)
=
4
IQR= 31-28
IQR= 3
2nd term
5th term
28
31
freq
27
1
28
2
30
1
31
1
33
1