– identifies the number of standard deviations a particular value is from
the mean of its distribution
Z = (x  µ) ¸ σ
■
x = the data value of interest
■
µ = the population mean
■
σ = the population standard deviation
○
Example: Calories in Hamburgers – Zscore for the WHOPPER
Hamburger Type
Restauran
t
Cal
ori
es
Cheeseburger
McDonald'
s
300
Single with
Everything
Wendy's
430
Big Mac
McDonald'
s
540
Whopper
Burger
King
670
Bacon
Cheeseburger
Sonic
780
Baconator
Wendy's
840
Triple Whopper with
Cheese
Burger
King
123
2/3 lb. Monster
Thickburger
Hardee's
142
Sample
Mean
776
.25
Standard
Deviation
385
X = 640
µ = 776.25
σ = 385.1
■
Z = (640 – 776.25) ¸ (385.1) = 0.35
○
ZScore Attributes
■
Positive for values above the mean
■
Negative for values below the mean
■
Zero for values equal to the mean
■
Has no units, even though the original values will normally be expressed
in other measurements

Analogous to expressing original data in other units
○
Outliers
– extreme values in the data set that require special consideration
■
Can be identified using the zscore
■
Within (+ / ) 3 = not considered outliers
■
A zscore that is 3.1 or 3.6 is considered an outlier
○
Calculating ZScores in Excel
■
=STANDARDIZE (x, mean, standard_deviation)
■
Example from above
X = 640
µ = 776.25
σ = 385.1
=STANDARDIZE (640, 776.3, 385.1)
●
The Empirical Rule
○
Empirical Rule
– if a distribution follows a bellshaped, symmetrical curve
centered around a mean, we would expect approximately 68%, 95%, and 99.7%
of the values to fall within one, two, and three standard deviations above and
below the mean, respectively
■
Almost all of the values will fall w/in 3 SD of the mean
■
AKA 689599.7 Rule
■
Example:

Average exam score is 80 points & SD is 5 points

Average PLUS and MINUS standard deviation (5)

à 68% of exam scores will fall b/w 75 and 85 points
○
Formula for Expressing the zScore in Terms of X
■
X = µ + zσ

X = average

µ = mean

z = standard deviation
○
68% fall within one standard deviation of the mean
■
Find data value that are one standard deviation above the mean

Set z = 1
■
Find data value that are one standard deviation below the mean

Set z = 1
○
95% fall within two standard deviations of the mean
■
Find data value that are two standard deviations above the mean

Set z = 2
■
Find data value that are two standard deviations below the mean

Set z = 2
○
99.7% fall within three standard deviations of the mean
■
Find data value that are three standard deviations above the mean

Set z = 3
■
Find data value that are three standard deviations below the mean

Set z = 3
●
Chebyshev’s Theorem
○
Chebyshev’s Theorem
– a mathematical rule that states that for any number z >
1, the percent of the values that fall within z standard deviations from the mean
will be AT LEAST = (11/z²) • (100)
■
Like empirical rule but applies to any distribution

Not just bellshaped, symmetrical distributions
■
**can only be stated for zscores that are greater than 1
■
Ex: when z = 2

[1 – (1/4)] = 0.75 (100) = 75%

à 75% of the data values will fall w/in 2 standard deviations of the mean (at least)
○
Conclusions
■
You've reached the end of your free preview.
Want to read all 94 pages?
 Fall '12
 Donnelly
 Standard Deviation, Mean