Algebra 1 Notes SOL A.9 Statistical Variation
Mr. Hannam
Name: ______________________________________ Date: _______________ Block: __________
Statistical Variation
Notation/Term
population
Description
An entire set of data about which
we wish to gain information.
sample
A subset of data from a population.
We use samples to make inferences
about populations.
xi
The ith element of data in a sample
or population.
mean of a population Represented by symbol µ (Greek
letter “mu”).
mean of a sample
deviation
Represented by x (x bar)
The distance a data point is from
the mean of the data. Can be
positive, negative, or zero.
Σ
The Greek uppercase symbol Σ
(sigma) is used to indicate
summation. The terms inside the
sigma are added together based on
an indexing scheme surrounding
the Σ
Mean of absolute values of the
deviations of elements from the
mean of a data set.
mean absolute
deviation
variance
standard deviation
The average of the squared
deviations from the mean. Variance
is represented by the lower case
symbol sigma squared (σ2)
The square root of the variance,
denoted by σ.
Example/Notes
The height of every middle
school student in the U.S.
The heights of students in our
class.
In data set {2, 4, 5, 10}, x2 = 4
For our purposes, we will use
x in place of µ in formulas.
Symbol our calculator uses.
Find by calculating x - µ or
xx
5
i 1 2  3  4  5 ;
i 1
4
x
i 1
Also called standard score; measure
of position that determines the
number of standard deviations an
element is above or below the mean
of a data set.
 x1  x 2  x 3  x 4
mean absolute deviation =
n
MAD =
x
i 1
i
μ
n
variance =
n
σ2 =
 x
i 1
 μ
2
i
n
standard deviation =
n
σ=
z-score
i
 x
i 1
 μ
2
i
n
z-score = z =
xμ
σ
Algebra 1 Notes SOL A.9 Statistical Variation
Mr. Hannam Page 2
Population vs. Sample

population: a well-defined group of objects about which we wish to gain information.

Examples: the heights of every U.S. President, the ages of every math teacher in VA.

sample: subset of a population. It is often very
o difficult to obtain data for every member of a population
o samples obtained randomly or by other methods; may or may not be biased

Different notations are used if referring to samples or populations.
o Ex: µ: mean of population; x : mean of sample
o WE WILL ALWAYS USE POPULATION NOTATION AND FORMULAS

We refer to a specific element, the ith element, of a sample or population as xi.
Measures of Spread or Variability

Measures of variability attempt to describe the spread of a set of data.

Some measures of variability we know: range and interquartile range.

New: average distance from the mean also measures spread
o Example: ages of children in a family (take your own notes on this!)
o A small average distance from the mean indicates data that is close together (alike)
o A large average distance from the mean indicates data that is farther apart (less
alike)
Mean Absolute Deviation (MAD): average of distances from the mean
o Deviation is the distance a point is from the mean:
o can be positive, negative, or zero
o found by subtracting the mean (µ) from the data point value
(x): x - µ.
o n = the number of points in a data set
o Absolute Deviation is the absolute value of deviation: | x - µ |
o We need to take absolute values of deviations so that the average isn’t zero
Example 1: S = {1, 3, 4, 6, 7, 9} µ = 5 (add the numbers and divide by 6)
Deviation = x - µ for each x
x
x-µ
|x-µ|
1
1 – 5 = -4
MAD _________
3
3 – 5 = -2
4
4 – 5 = -1
6
6–5=1
7
7–5=2
9
9–5=4
Algebra 1 Notes SOL A.9 Statistical Variation
Mr. Hannam Page 3
Example 2: Find the MAD for the dataset in the table
Mean (µ) = _________
Distances:_______________________________________________________________
MAD _________
You Try: Find the MAD for the dataset S = {11, 29, 40, 50, 60, 65, 95}
Mean (µ) = _________
n (number of points) = _________
Distances:_______________________________________________________________
MAD _________
Summation Notation
The Greek uppercase letter Σ (sigma) is used to denote
summation (adding).
Examples:
5
 i  1  2  3  4  5  15
i 1
n
x
i 1
i
 x 1  x 2  x 3  ...x n for S  {4 2 8 10 1 12}
Example above…
 Traverses data
 n = number of data points
 Subscript notation is positional; NOT AN EXPONENT!!
 xi refers to the ith element
 Example: x3 refers to the third element; above it refers to the value 8
 Above, the summation is asking as to sum all the data in the data set. Evaluate the
summation. __________
You try: Evaluate the summations…
4
3
5
n
e) Rewrite the formula for
a)  2i
b)  i 2
c)  (i  1)
d)  x i
mean using Σ notation.
i 1
i 1
i 1
i 1
for {8 4 6 9}
Re-write the MAD formula using summation notation:
n
Mean Absolute Deviation (MAD) 
x
i 1
i
n
μ
Algebra 1 Notes SOL A.9 Statistical Variation
MAD =
sum of x - μ
n
Mr. Hannam Page 4
becomes
Standard Deviation
Another way to ensure that deviations are always positive is to square them.

We can sum the squares of the deviations and divide by the number of deviations to get
variance, denoted by σ2:
n
x i  μ 2

Variance = σ2 = i 1
n

The square root of variance is a closer approximation of the deviation of points from the
center. We call the square root of variance, σ, standard deviation.
n
Standard Deviation = σ =
 x
i 1
 μ
2
i
n
Example 1 revisited: S = {1, 3, 4, 6, 7, 9}, µ = 5.

Find the variance and standard deviation are as follows:
x
x-µ
(x - µ )2
1
1 – 5 = -4
3
3 – 5 = -2
4
4 – 5 = -1
6
6–5=1
7
7–5=2
9
9–5=4
variance = σ2 = _____________ standard deviation = σ = __________ (square root of variance)
You try: Find the MAD, variance, and standard deviation of S = {1, 1, 3, 4, 5}
Example: Find the standard deviation for the ages of kids in a family
{0, 10, 11, 14, 15, 20}, µ = 12 (approximately)
x
x-µ
0
10
11
14
15
20
(x - µ )2
variance = σ2 = _____________ standard deviation = σ = __________
We previously calculated the MAD for this data: MAD = 4.7

How do the MAD and standard deviations for the data compare?_____________

Why do you think this is so? _______________________________________________
Using the Calculator
It’s a lot of work to find these descriptive statistics by hand! Fortunately, the calculator
comes to our rescue. Use the calculator guide handout to find the mean absolute deviation
(MAD), variance, and standard deviation of the following data sets:
Algebra 1 Notes SOL A.9 Statistical Variation
a) {10, 20, 30, 40, 50, 60}
Mr. Hannam Page 5
b) {10, 12, 7, 11, 20, 7, 6, 8, 9}
c) {1202, 1229, 1012, 1014,
1120, 1429}
Z-Scores (Standard Scores)

We use standard deviations to assess average distance from the mean for a set of data.

A z-score tells us how many standard deviations an element is above or below the
mean of the data set.
Standard Score = z-Score = z =
xμ
σ

positive z-score: the point is that many standard deviations above the mean

negative z-score: the point is that many standard deviations below the mean
Example: A set of values has a mean of 85 and a standard deviation of 6. Find the z-score
of the value 76.
z-score =
x  μ 76  85  9


 1.5
σ
6
6
This tells us that 76 is 1.5 standard deviations below the mean.
Example: The following numbers are metabolic rates of 7 members who took part in a
study of dieting (units are calories per 24 hours - these are the same calories used to
describe the energy content of foods).
1792 1666 1362 1614 1460 1867 1439
How many standard deviations above or below the mean is 1460 calories?

µ = ____________ σ = ____________ z-score ___________

How many points above are within 2 standard deviations from the mean? __________

How many points are within 1.5 standard deviations from the mean? __________
Example: John weighs 220 lbs; his dog Fido weighs 90 lbs. If human males weigh an
average of 160 lbs with a standard deviation of 20 lbs, and all dogs of Fido’s breed have an
average weight of 80 lbs with a standard deviation of 5 lbs, how do John and Fido compare,
relative to their populations, with respect to weight?
John’s z-score =
Fido’s z-score =
Conclusion?
Algebra 1 Notes SOL A.9 Statistical Variation
Mr. Hannam Page 6
Example: Given a set of data with a standard deviation of 3.0 and a mean of 9.0, what
would the value of an element be if its z-score is 2.5?
You try:
a) Find the z-score for a data value of 15 if a set of data has a mean of 75, and a standard
deviation of 5.
b) How many standard deviations from the mean is 80 if a set of data has a mean of 70
and a standard deviation of 2.5?
c) Find the value of an element in a dataset with a mean of 8, a standard deviation of 2,
and a z-score of 1.5.
d) Test A has a mean of 50 and σ = 10. Test B has a mean of 20 and σ = 5. Which is
better: a score of 65 on test A or a score of 29 on test B?