3. Square the differences from step 2 à new column
-
We do this because the sum of the differences column will always be zero
■
4. Add up the total of the squared values
-
Ex: (0 + 625 + 225 + 225 + 625) = 1,700
-
AKA sum of squares
■
5. Divide sum of squares (1700) by (n-1)
-
Where n = number of data points in beginning
-
Ex: (1700) ¸ (5-1) =
425
-
Technically, the result is in units squared
○
Standard Deviation
– the square root of the variance
Formula: Ö variance
■
The unit of measure for the standard deviation is the same as the unit for
the original data
■
Measures the average distance b/w all of a set’s data points and the
mean
■
With variance, measuring how far on average each data value is from the
mean of the sample
○
Other Notes
■
The standard deviation and variance should NEVER be negative values
(this would be a mistake)
●
Short-Cut Formulas for the Sample Variance and Standard Deviation
○
Step-by-step
35
10
50
20
60
■
1. Sum the data values
-
Ex: (35 + 10 + 50 + 20 + 60) = 175
■
2. Square the result found in step 1
-
Ex: (175)² = 30,625
■
3. Square each of the original data values

■
4. Sum the squared column you created in step 3
-
Ex: (1225 + 100 + 2500 + 400 + 3600) = 7,825
■
5. [(4 answer) – (2 answer ¸ # data points)] ¸ (n-1)
-
Ex: (7825 - 30,625/5) ¸ (4) = 425
●
The Variance and Standard Deviation for a Population
○
The Population Variance
■
Instead of (n-1) on the bottom of the equation, just use N
■
N = total number of data points in the population
○
Population Standard Deviation
■
Square root of population variance
●
Using Excel to Calculate the Variance and Standard Deviation
○
The Sample Functions
■
=VAR (data values)
■
=STDEV (data values)
○
Finding Sample Variability
■
Data Analysis
(add-in) >
Descriptive Statistics
■
Select input range for all data points
■
Output Range (random selection)
■
Summary Statistics
> gives standard deviation, sample variance, and
range values
○
The Population Functions
■
=VARP (data values)
■
=STDEVP (data values)
Ø
3.3 – USING THE MEAN AND STANDARD DEVIATION TOGETHERS
o
Some Notes
§
When means are similar / same, compare their standard deviations
§
When sample means are very different, comparing standard deviations can be misleading
-
b/c size of standard deviation is affected by the scale of the data
●
The Coefficient of Variation
○
Coefficient of Variation
– (CV) measures the standard deviation in terms of its
percentage of the mean
■
High CV = high variability, less consistency
■
Low CV = low variability, more consistency
■
Best when comparing consistency b/w two data sets when their means
are very different
-
b/c 2 data sets are likely to have diff means

■
Gives a common measure – percentage – to compare 2+ things
■
Measures the standard deviation in terms of its percentage of the mean
○
Formula for the Sample Coefficient of Variation
CV = S ¸ X (100)
■
S = the sample standard deviation
■
X = the sample mean
○
Formula for the Population Coefficient of Variation
CB = σ ¸ µ
■
σ = the population standard deviation
■
µ = the population mean
●
The z-Score
○
Z-Score


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- Fall '12
- Donnelly
- Standard Deviation, Mean