To find the standardized score (z-score) for a piece of data:
An
example using the data set {3, 5, 7, 8, 12, 13, 15, 16, 16, 18, 19, 24}
(This example includes the steps for finding a standard deviation).
Find the mean of your data by adding
all the data values and dividing by 12 (there are 12 pieces of data)
3 + 5 + 7+ 8+ 12+ 13+ 15+ 16+ 16+ 18+ 19+ 24 = 156
156/12 = 13
Subtract the mean from each piece of
data to find the deviations
3 - 13 = -10
5 - 13 = -8
7 - 13 = -6
8 - 13 = -5
12 - 13 = -1
13 - 13 = 0
15 - 13 = 2
16 - 13 = 3
16 - 13 = 3
18 - 13 = 5
19 - 13 = 6
24 - 13 = 11
Square each of the deviations
(-10)^2 = 100
(-8)^2 = 64
(-6)^2 = 36
(-5)^2 = 25
(-1)^2 = 1
(0)^2 = 0
(2)^2 = 4
(3)^2 = 9
(3)^2 = 9
(5)^2 = 25
(6)^2 = 36
(11)^2 = 121
Find the mean of the squared
deviations (this is the variance)
100 + 64 + 36 + 25 + 1 + 0 + 4 + 9 + 9 + 25 + 36 + 121 = 430
430/12 = 35.833... = variance
Find the square root of the mean of
the squared deviations (this is the standard deviation)
SQRT(35.833 ...) = 5.98609... = standard deviation
To find the standardized score (z-score) for a piece of data with a calculator:
Clear your previous lists (L1, L2, L3, etc.)
STAT » EDIT » 4:ClrList » 2^{nd} » 1 » , » 2^{nd} » 2 » , » 2^{nd} » 3 » ENTER
Enter all of the data into list one
STAT » EDIT » 1:EDIT
Find the mean of list one
STAT » CALC » 1:1Var Stats » 2^{nd} » 1 » ENTER
The mean is the first number listed (X bar)
Subtract the mean from each piece of data to find the deviations
STAT » EDIT » 1:EDIT
Move to the right and up so your cursor is on L2
Write the formula "L1 - X bar"
ALPHA » + » 2^{nd} » 1 » - » VARS » 5:STATISTICS » 2:Xbar »ALPHA» + » ENTER
Divide the deviation by the
standardized deviation
Move to the right and up so your cursor is on L3
Write the formula "L2/σ x "
ALPHA » + » 2^{nd} » 2 » / » VARS » 5:STATISTICS » 4:σ x »ALPHA» + » ENTER
The result is the standardized score (z-score)
Z-scores show how many standard
deviations away from the mean a piece
of data is.