#### Standard Deviation & Z-Scores

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Standard Deviation & Z-Scores

Sigma (Summation) Notation
Consider the heights of the Jones family
members:
Mom 65” Dad 72”
Bobby 70” Joey 58”
Fluffy 22” Spot 28”
Let’s find the mean (m - “mu”).
65 + 72 + 70 + 58 + 22 + 28 = 315
m = 327 ÷ 6 = 52.5”
Let’s look at those heights as a “set”.
{65, 72, 70, 58, 22, 28}
{x1, x2, x3, x4, x5, x6}
To find the mean, we added the
elements of the set
65 + 72 + 70 + 58 + 22 + 28
x1 + x2 + x3 + x4 + x5 + x6
Summation notation shows it this way:
𝑛
𝑥𝑖 where n = 6
𝑖=1
S Is the capital Greek letter “sigma”
Mean (m) in Summation Notation
m=
1.
2.
𝑛
𝑥
𝑖=1 𝑖
𝑛
We are still calculating the mean
(average or “mu”) the same as always…
Sum the elements of the set
Divide by the number of elements.
Deviation – how far is a certain
piece of data from the mean?
So here is our family…
Here is the mean of 52.5”
that we calculated earlier.
Since no single family member is exactly 52.5”
tall, all of them deviate from the mean. Some
more than others…
Mean Absolute Deviation &
Standard Deviation
These are 2 different statistics to
spread (dispersion) of
describe the ________
the data.
Mean Absolute Deviation (MAD) – is
often preferred because it is less
affected by __________.
outliers
Standard Deviation (s) - A more
traditional way of indicating dispersion.
Calculating Mean Absolute
Deviation (MAD)
MAD =
𝑛
|𝑥 −𝜇|
𝑖=1 𝑖
𝑛
Step 1: Find the mean m of the data.
52.5”
Step 2: Subtract the mean from each data element.
65-52.5”
12.5
72-52.5
19.5
70-52.5
17.5
58-52.5
5.5
22-52.5
-30.5
28-52.5
-24.5
Calculating Mean Absolute
𝑛
Deviation (MAD)
|𝑥 −𝜇|
𝑖=1 𝑖
MAD =
12.5
19.5
17.5
5.5
𝑛
-30.5
-24.5
Step 3: Take the absolute value of the values found
in Step 2
12.5
19.5
17.5
5.5
30.5
24.5
Step 4: Sum those numbers.
12.5 + 19.5 + 17.5 + 5.5 + 30.5 + 24.5 = 110
Step 5: Divide by “n” (the number of values) 110 ÷ 6
MAD = 18.3
Now that we have it, what is it
for?
We can use the MAD (18.3), to determine
which of our data (family members) are
unusually tall or unusually small.
The mean was 52.5”, so anyone
taller than: 52.5 + 18.3 70.8”
or smaller than : 52.5 – 18.3 34.2”
is “unusual”.
So, Dad (72”), Fluffy (22”) and Spot (28”) have
unusual heights for the Jones family.