Standard Deviation & Choosing Measures of Center and Spread
Example: # of pets: 1 3 4 4 4 5 7 8 9
Mean = 𝑥̅ = 5
*Challenge: come up with a number that measures how far this data is from the mean
data
1
3
4
4
4
5
7
8
9
TOTAL:
Data - 𝑥̅
1 – 5 = -4
3 – 5 = -2
-1
-1
-1
0
2
3
4
0
Can’t have negative distances! What do we do?
(data - 𝑥̅ )2
16
4
1
1
1
0
4
9
16
52
Squaring makes it positive
*Variance: the average of the squares of
the deviations of the observations from
their mean. (s2)
𝑠2 =
Should always add up to be zero
Variance:
Because we do not want out units squared,
we must take the square root. Giving us the
standard deviation.
52
1
∑(𝑥𝑖 − 𝑥̅ )2
𝑛−1
=
9−1
52
8
≈ 6.5 ≈ s2
*Standard Deviation: the square root of
the variance. (s)
Standard deviation is an average measure of
how far the data is from the mean.
1
𝑠=√
∑(𝑥𝑖 − 𝑥̅ )2
𝑛−1
Standard deviation: √6.5 ≈ 2.5495 ≈ 𝑠
Properties of standard deviation:
 s measures spread about the mean and should only be used with mean (not with median)
 s = 0 only when there is no spread. This means all the observations are the same.
 s, like the mean, is not resistant. Outliers do influence it.
SO WHAT IS BEST TO USE?
Center
Non Resistant
Mean, 𝑥̅
Resistant
Median, M
Homework: p. 69 # 79, 81, 89, 98 a & b, 100
Spread
Standard deviation, s
IQR
Due: Block Day
Use for distributions that
are relatively symmetric,
no outliers
Use for strongly skewed
distributions, outliers