A Look at Means, Variances,
Standard Deviations, and z-Scores
Dr. Margie Mason
College of William and Mary
mmmaso@wm.edu
Based on the
Technical Assistance Document
2009 Algebra I Standard of
Learning A.9
http://www.doe.virginia.gov/instruction/
high_school/mathematics/technical_assis
tance_algebra1_a9.pdf
Algebra Standard of Learning A.9
The student, given a set of data, will
interpret variation in real-world
contexts and calculate and interpret
mean absolute deviation, standard
deviation, and z-scores.
Mathematics SOL 5.16
5.16 The student will
a) describe mean, median, and
mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode,
and range of a set of data; and
d) describe the range of a set of data
as a measure of variation.
Algebra SOL A.10
The student will compare and
contrast multiple univariate data
sets, using box-and-whisker plots.
Algebra SOL A.10
Let’s use unifix stick heights of
5
13
6
15
8 8 10
17 18 20
Determining
a Box-and-Whisker Plot
1. Find:
Median – the middle value when arranged from
smallest to largest. 11.5
Lower extreme (LE) – the smallest value 5
Upper extreme (UE) – the largest value 20
Lower quartile (LQ) – the value halfway between
the lower extreme and the median. 8
Upper quartile (UQ) – the value halfway between
the upper extreme and the median. 17
Interquartile range (IQR) – The difference between
the upper quartile and the lower quartile. 9
Determining
a Box-and-Whisker Plot
2. Determine and draw the scale.
Subtract the smallest value from the largest
value to determine the range. Choose a
reasonable size for the intervals based on the
range to be covered, e.g., 1, 2, 5, or 10.
Draw the scale much like a number line at
the bottom of your plot.
Determining
a Box-and-Whisker Plot
3. Draw the box:
a. Length of the box extends for LQ to
UQ. It is drawn above the scale.
b. Mark the median.
c. Width of box can be anything.
Determining
a Box-and-Whisker Plot
4. Draw the whiskers:
a. Draw from the box you just drew to
LE and UE.
Determining
a Box-and-Whisker Plot
Alternate method for drawing the
whiskers:
4. Determine the outliers: Multiply the
IQR by 1.5 and add this number to the
UQ and subtract it from the LQ. Any
values outside these limits are outliers.
Determining
a Box-and-Whisker Plot
Alternate Method:
5. Draw whiskers:
a. Draw from box to LE and UE
excluding outliers.
b. Place asterisks on any outliers.
Determining
a Box-and-Whisker Plot
How many years of experience as a
teacher do you have?
Mathematics SOL 6.15
The student will
a) describe mean as balance point; and
b) decide which measure of center is
appropriate for a given purpose.
Mathematics SOL 6.15
The mean is the numerical average of the
data set and is found by adding the
numbers in the data set together and
dividing the sum by the number of data
pieces in the set.
In grade 5 mathematics, mean is defined
as fair-share.
Mathematics SOL 6.15
Mean can be defined as the point on a number
line where the data distribution is balanced. This
means that the sum of the distances from the
mean of all the points above the mean is equal to
the sum of the distances of all the data points
below the mean. This is the concept of mean as
the balance point.
Defining mean as balance point is a prerequisite
for understanding standard deviation.
Mathematics SOL 6.15
Balance Point: The sum of the distances on a
number line from the mean of all the points
above the mean is equal to the sum of the
distances of all the data points below the
mean.
7+6+4+4+2=1+3+5+6+8
Sample vs. Population Data
A statistical population includes all elements
in the set.
A sample is a subset of the population.
A data set, whether a sample or population,
is comprised of individual data points
referred to as elements of the data set.
Start with small defined population data sets
of approximately 30 items or less.
*Elements*
An element of a data set will be
represented as xi. Where i represents the
ith term of the data set.
Sample vs. Population Data
The arithmetic mean of a population is
represented by the Greek letter m (mu),
while the calculated arithmetic
mean of a sample is represented by
, read “x bar.”
Mean Absolute Deviation vs.
Variance and Standard Deviation
Measuring dispersion or spread of a data
set about the mean
One measure of spread is to find the sum
of the deviations between each element
and the mean; however, this sum is
always zero.
Mean Absolute Deviation vs.
Variance and Standard Deviation
Two methods:
take the absolute value of the deviations
before finding the average, (Mean
Absolute Deviation)
or
square the deviations before find the
average (Variance and Standard
Deviation)
Mean Absolute Deviation vs.
Variance and Standard Deviation
Summation Notation
Mean Absolute Deviation
The arithmetic mean of the absolute values of the deviations
of elements from the mean of a data set.
5 6 8 8 10 13 15 17 18 20
|5-12| + |6-12| + |8-12| + |8-12| +|10-12| + |13-12| + |15-12| +|17-12| + |18-12| + |20-12|
10
=7+6+4+4+2+1+3+5+6+8
10
= 46
10
= 4.6
*Mean Absolute Deviation*
Mean absolute deviation =
where m represents the mean of the data
set,
n represents the number of elements in the
data set, and
xi represents the ith element of the data set.
Mean Absolute Deviation
Mean absolute deviation is less affected
by outlier data than the variance and
standard deviation. Outliers are elements
that fall at least 1.5 times the interquartile
range (IQR) below the first quartile (Q1) or
above the third quartile (Q3).
Variance
The average of the squared deviations from the
mean is known as the variance and is another
measure of the spread of the elements in the set.
(5-12)2 + (6-12)2 + (8-12)2 + (8-12)2 +(10-12)2 + (13-12)2 + (15-12)2 +(17-12)2 + (18-12)2 + (20-12)2
10
= (-7)2 + (-6) 2 + (-4) 2 + (-4) 2 + (-2) 2 + 12 + 32 + 52 + 62 + 82
10
= 49 + 36 + 16 + 16 + 4 + 1 + 9 + 25 + 36 + 64 = 256 = 25.6
10
10
*Variance*
Variance (s 2)=
where
m represents the mean of the data set,
n represents number of elements in the
data set, and
xi represents the ith element of the data
set.
Variance
The differences are squares so that they don’t
cancel each other out when finding the sum.
When squaring the differences, the units of
measure are squared and the larger differences
are “weighted” more heavily than smaller
differences.
In order to provide a measure of variation in
terms of the original units of the date, the
square root of the variance is taken, yielding
the standard deviation.
Standard Deviation
The positive square root of the variance
of the data set.
The greater the value, the more spread
out the data are about the mean.
The lesser (closer to 0) the value, the
closer the data are clustered about the
mean.
*Standard Deviation*
Standard deviation (s)=
where
m represents the mean of the data set,
n represents the number of elements in
the data set, and
xi represents the ith element of the data
set.
Standard Deviation
“ s ”, written and read “sigma”,
represents the standard deviation of a
population and “s” represents the
sample standard deviation.
s = the square root of 25.6
= 5.06
Standard Deviation
The population standard deviation can be
estimated by calculating the sample
standard deviation. The formulas for
sample and population look similar except
that the sample standard deviation formula
uses n – 1 instead of n in the denominator.
This is to account for the possibility of
greater variability of data in the population
than what was seen in the sample.
Standard Deviation
When n-1 is used in the denominator, the
result is a larger number. So the calculated
value of the sample standard deviation will
be larger than the population standard
deviation. As sample sizes get larger, the
difference gets smaller. The use of n-1 is
known as Bessel’s correction. SOL A.9 used
the population standard deviation with n in
the denominator.
Interpreting Standard Deviation
•
•
Standard deviation is a measure of the typical
amount an entry deviates from the mean.
The more the entries are spread out, the greater
the standard deviation.
Interpreting Standard Deviation
Empirical Rule (68 -95-99.7 Rule)
For data with a (symmetric) bell-shaped
distribution, the standard deviation has the
following characteristics:
•About 68% of the data lie within one
standard deviation of the mean.
•About 95% of the data lie within two
standard deviations of the mean.
•About 99.7% of the data lie within three
standard deviations of the mean.
Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
95% within 2 standard deviations
68% within 1
standard deviation
34%
2.35%
x  3s
34%
13.5%
x  2s
13.5%
x s
x
x s
2.35%
x  2s
x  3s
Example: Using the Empirical Rule
In a survey conducted by the National
Center for Health Statistics, the sample
mean height of women in the United
States (ages 20-29) was 64.3 inches,
with a sample standard deviation of 2.62
inches. Estimate the percent of the
women whose heights are between 59.06
inches and 64.3 inches.
Example: Using the Empirical Rule
• Because the distribution is bell-shaped, you can
use the Empirical Rule.
34% + 13.5% = 47.5% of women are
between 59.06 and 64.3 inches tall.
Chebychev’s Theorem
•
The portion of any data set lying within k standard
1
deviations (k > 1) of the mean is at least:
1
k
•
2
k = 2: In any data set, at least 1  12  3 or 75%
2
4
of the data lie within 2 standard deviations of the
mean.
1
8
• k = 3: In any data set, at least 1  3  9 or 88.9%
of the data lie within 3 standard deviations of the
mean.
2
Using Chebychev’s Theorem
The age distribution for Florida is shown in the
histogram. Apply Chebychev’s Theorem to the
data using k = 2. What can you conclude?
Using Chebychev’s Theorem
k = 2: μ – 2σ = 39.2 – 2(24.8) = – 10.4 (use 0 since
age can’t be negative)
μ + 2σ = 39.2 + 2(24.8) = 88.8
At least 75% of the population of Florida is
between 0 and 88.8 years old.
z-Scores
A z-score, also called a standard score, is a
measure of the position derived from the mean
and standard deviation of the data set.
In Algebra I, the z-score will be used to
determine how many standard deviations an
element is above of below the mean of the data
set. It can also be used to determine the value
of the element, given the z-score of an
unknown element and the mean and standard
deviation of a data set.
The Standard Score (z-Score)
Represents the number of standard
deviations a given value x falls from the
mean μ.
value - mean
x-m
z=
=
standard deviation
s
z-Scores
The z-score will be positive if the
element lies above the mean and
negative if it lies below the mean. A
z-score is calculated by subtracting the
mean of the data set from the element
and dividing the result by the standard
deviation of the data set.
*z-Scores*
z-score (z) =
where x represents an element of the
data set, m represents the mean of the
data set, and s represents the standard
deviation of the data set.
z-Scores
z-score of 5 = (5 - 12)/5.06 = -1.38
z-score of 6 = (6 - 12)/5.06 = -1.19
z-score of 8 = (8 - 12)/5.06 = -.79
z-score of 10 = (10 - 12)/5.06 = -.40
z-score of 13 = (13 - 12)/5.06 = .20
z-score of 15 = (15 - 12)/5.06 = .59
z-score of 17 = (17 - 12)/5.06 = .99
z-score of 18 = (18 - 12)/5.06 = 1.19
z-score of 20 = (20 - 12)/5.06 = 1.58
z-Scores
Suppose you had the misfortune to have an Algebra
test and a history test on the same day. Why you got
your tests back, here is the information given to you
regarding your performance and the performance of
the class on these exams. Which test did you do
better on?
Algebra
History
Your score
82
93
Mean
71.06
85.43
Stan. Dev
10.32
18.91
z-Scores
Suppose the mean and standard deviation on an
algebra test were given as 72 and 12, respectively.
Susan’s z-score on the test was 2.34. Was Susan’s
test score above or below the mean? How do you
know?
David’s z-score on the same test was -1.25. Was
David’s test score above or below the mean? How
do you know?
z-Scores
Suppose the mean and standard deviation on
an algebra test were given as 72 and 12,
respectively.
Dakota had a z-score of 0.08 on the test.
What does this z-score tell you about Dakota’s
test score relative to the mean?
z-Scores
Suppose the mean and standard deviation on
an algebra test were given as 72 and 12,
respectively.
If the z-score for Susan was 2.34, for David
was -1.25, and for Dakota was 0.08, find their
actual test scores.
z-Scores
Suppose the mean and standard deviation on
an algebra test were given as 72 and 12,
respectively.
Rebecca made an 80 on the biology test. Find
her z-score.
Interpretation of
Descriptive Statistics
Data set 1
Number of Basketball Players Recorded
Once Each Day from April 1-14
Data set 2
Number of Basketball Players Recorded
Once Each Day from April 15-28
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1-10 11-20 21-30 31-40 41-50 51-60 61-70
Number of Players
Mean = 45.0
Variance = 106.1
Standard Deviation = 10.3
Mean Absolute Deviation = 9.1
1-10 11-20 21-30 31-40 41-50 51-60 61-70
Number of Players
Mean = 45.0
Variance = 420.3
Standard Deviation = 20.5
Mean Absolute Deviation = 16
Interpretation of
Descriptive Statistics
Maya represented the heights of boys in Mrs. Constantine’s and Mr. Kluge’s classes on a line plot
and calculated the mean and standard deviation.
Heights of Boys in Mrs. Constantine’s
and Mr. Kluge’s Classes (in inches)
x
64
x
65
x
x
x
66
x
x
67
x
x
x
x
x
68
x
x
x
69
x
x
70
x
71
x
x
72
Mean = 68.4 Standard Deviation = 2.3
Note: In this problem, a small, defined population of the boys in Mrs. Constantine’s and Mr.
Kluge’s classes is assumed.
x
73