2.4 measures of variation
DESCRIPTION
2.4 Measures of Variation. The Range of a data set is simply: Range = (Max. entry) – (Min. entry). Deviation. The deviation of an entry, x , is the difference between the entry and the mean, , of the data set. Mean = Deviation of x = x - . Population Variance. - PowerPoint PPT PresentationTRANSCRIPT
2.4 Measures of Variation
The Range of a data set is simply:Range = (Max. entry) – (Min. entry)
Deviation
• The deviation of an entry, x, is the difference between the entry and the mean, , of the data set.
• Mean = • Deviation of x = x -
N
x
Population Variance• We are not going to be talking much about the
Population Variance. We will be talking more about the Sample Variance.
• Population Variance is found by:– Find the mean of the population (note the symbol)– Find the deviation of each point by subtracting the
mean from each data point– Square the differences– Add all the squares up– Divide by the total number of data points in the
population• Population Variance:
N
x 22
Population Standard Deviation
• The Population Standard Deviation is the square root of the Population Variance.
N
x 22
Sample Variance• We will be talking mostly about the Sample
Variance. • Why?• Sample Variance is found by:– Find the mean of the sample:– Find the deviation by subtracting the mean of the
sample from each data point– Square the differences– Add all the squares up– Divide by the total number of data points in the
sample minus 1.• Sample Variance:
n
xx
1
2
2
n
xxs
Sample Standard Deviation
• The Sample Standard Deviation is the square root of the Sample Variance.
1
2
2
n
xxss
Example• Find the standard deviation of the following
sample:
X
5
7
6
6
5
9
8
8
Sum
Example• Find the standard deviation of the following
sample:
X
5
7
6
6
5
9
8
8
Sum
1
2
2
n
xxss
Example• Find the standard deviation of the following
sample:
X
5
7
6
6
5
9
8
8
Sum
n
xx
1
2
2
n
xxss
Example• Find the standard deviation of the following
sample:
X
5
7
6
6
5
9
8
8
Sum 54
75.68
54
n
xx
1
2
2
n
xxss
Example• Find the standard deviation of the following
sample:
X
5 -1.75
7 0.25
6 -0.75
6 -0.75
5 -1.75
9 2.25
8 1.25
8 1.25
Sum 54
xx 75.6
8
54
n
xx
1
2
2
n
xxss
What will be the sum of this column?
Example• Find the standard deviation of the following
sample:
X
5 -1.75
7 0.25
6 -0.75
6 -0.75
5 -1.75
9 2.25
8 1.25
8 1.25
Sum 54 0
xx 75.6
8
54
n
xx
1
2
2
n
xxss
What will be the sum of this column?
It will always be zero
Example• Find the standard deviation of the following
sample: 1
2
2
n
xxss
X
5 -1.75 3.0625
7 0.25 0.0625
6 -0.75 0.5625
6 -0.75 0.5625
5 -1.75 3.0625
9 2.25 5.0625
8 1.25 1.5625
8 1.25 1.5625
Sum 54
xx 2xx
75.68
54
n
xx
Example• Find the standard deviation of the following
sample: 1
2
2
n
xxss
X
5 -1.75 3.0625
7 0.25 0.0625
6 -0.75 0.5625
6 -0.75 0.5625
5 -1.75 3.0625
9 2.25 5.0625
8 1.25 1.5625
8 1.25 1.5625
Sum 54 15.5
xx 2xx
75.68
54
n
xx
488.1
18
5.152
ss
Standard Deviation
• The TI calculators can calculate both standard deviations quickly:– Stats– Calc– 1-Var Stats– Enter the list you want to use– Enter
Standard Deviation
• This gives:– The mean of the data: – The sum of all of the data: – The sum of the squares of all the data: – Sample standard deviation: – Population standard deviation: – The number of data points: – The smallest data point value: minX– Etc.
xx
2xSx
xn
Standard Deviation
• What does Standard Deviation represent?• It is a measure of the distance from the mean.• It is a measure of how far the data is from the
mean.• It is a measure of the spread of data.• The larger the Standard Deviation, the more
spread out the data is.
Standard Deviation
• Calculate the mean, range, and standard deviations for 8 units at a value of 7:– Mean = 7– Range = 0– Population and Sample Standard Deviations = 0,
why?– There is no spread in the data. It is all the exact
same number
Standard Deviation
• Calculate the mean, range, and standard deviations for 4 units each at 6 and 8:– Mean = 7– Range = 2– Population Standard deviation = 1, why?– The data is an average of one unit from the mean– Sample Standard Deviation = 1.069, why?– We are dividing by (n-1)
Standard Deviation• Calculate the mean, range, and sample
standard deviation for 2 units each at 4, 6, 8 and 10:– Mean = 7– Range = 6– Sample Standard deviation = 2.39 and
Population Standard Deviation = 2.236, why not 2?– Even though the data is an average of 2 units from
the mean, the standard deviations are not exactly 2 because we are working with the square of the distances.
Standard Deviation Summary
• Standard deviation is the square root of variance• Population standard deviation has an “n” in the
denominator• Sample standard deviation has an “n – 1” in the
denominator• Both standard deviations is a measure of the
spread of data• The more the spread, the larger the standard
deviation
Standard Deviation in a Normal Curvefrom http://allpsych.com/researchmethods/images/normalcurve.gif
Standard Deviation in a Normal Curvefrom http://www.comfsm.fm/~dleeling/statistics/normal_curve_diff_sx.gif
Class Work
• Pg 79, # 16, 18, 24
Homework
• Page 78, # –5 – 9 all, –13 – 21 odd, –22–25 & 26 – Total of 13 problems