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Transcript
Variability
 Statistics means never having to say you're certain.
Statistics - Chapter 4
2
Variability
 The amount by which scores are dispersed or scattered
in a distribution.
 Page 74 graphs
Statistics - Chapter 4
3
Range
Difference between the largest and smallest scores.
Problem: large groups may have large range
Variance and Standard Deviation
 Standard Deviation - The square root of the variance.
 Or
 The square root of the mean of all the squared
deviations from the mean!!???
 The value of a standard deviation can NOT be
negative.
Standard Deviation
 A rough measure of the average (or standard) amount
by which scores deviate on either side of their mean.
Progress Check 4.1 Page 80
 Employees of Corporation A earn annual salaries
described by a mean of $90,000 and a standard
deviation of $10,000.
 a. The majority of all salaries fall between what two
values?
 b. A small minority of salaries are less than what value?
 c. A small minority of all salaries are more than what
value?
 c. Answer parts (a), (b), and (c) for Corporation B’s
employees, who earn annual salaries described by a
mean of $90,000 and a standard deviation of $2,000.
Standard Deviation
 Deviations from the mean.
 The sum of all the deviations equals the variance.
 To calculate the variance
 The sum of squares equals the sum of all squared
deviation scores (p. 83)
Sum of Squares
 Calculation example of sample sum of squares (SS)
using the computation formula (p. 83)
 SS=ΣX2 – (ΣX)2
n
Standard Deviation
 Calculation example of sample standard deviation
using the computation formula (p. 86)
 s = √s2 =
√
SS2
n-1
Why n-1? (p88)
 This applies the sample estimate to the variance rather
then the population estimate.
 If we use the population estimate we would
underestimate the variability.
 In other words, this allows a more conservative and
accurate estimate of the variance within the sample.
Degrees of freedom
 Degrees of freedom (df) refers to the number of values
that are free to vary, given one or more mathematical
restrictions, in a sample being used to estimate a
population characteristic. (p. 90)
The value of the population mean – mu (μ)
 Most of the time the population mean is unknown so
we use the value of the sample mean and the degrees
of freedom (df) = n-1.
Standard Deviation calculation (p88)
1.
2.
3.
4.
5.
6.
7.
8.
Assign a value to n representing the number of X scores.
Sum all X scores.
Square the sum of all X scores.
Square each X score.
Sum all squared X scores.
Substitute numbers into the formula to obtain the sum of
squares, SS.
Substitute numbers in the formula to obtain the sample
variance, s2.
Take the square root of s2 to obtain the sample standard
deviation, s.
Qualitative data and variance
 No measures of variability exist for qualitative data!
 However, if the data can be ordered, then the
variability can be described by identifying extreme
scores (ranks).
Progress Check
 Calculate the mean, median, mode, and standard
deviation for the following height of students in
inches.
 64, 61, 73, 70, 71, 75, 69, 60, 63, 71, 65, 62
Statistics - Chapter 4
16