Download Measuring Spread: The Standard Deviation

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Chapter 1: Exploring Data
Section 1.3
Describing Quantitative Data with Numbers
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
Spread: The Standard Deviation
The standard deviation sx measures the average distance of the
observations from their mean. It is calculated by finding an average of
the squared distances and then taking the square root. This average
squared distance is called the variance.
(x1  x ) 2  (x 2  x ) 2  ... (x n  x ) 2
1
variance = s 

(x i  x ) 2

n 1
n 1
2
x
1
2
standard deviation = sx 
(x

x
)

i
n 1
Describing Quantitative Data
Definition:
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 Measuring
Consider the following data on the number of pets owned by
a group of 9 children.
1) Calculate the mean.
2) Calculate each deviation.
deviation = observation – mean
deviation: 1 - 5 = -4
deviation: 8 - 5 = 3
x =5
Describing Quantitative Data

Spread: The Standard Deviation
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 Measuring
Spread: The Standard Deviation
4) Sum the squared deviations.
Then, divide by (n-1)…this is
called the variance (the
“average” squared deviation).
5) Take the square root of the
variance…this is the standard
deviation.
(xi-mean)
1
1 - 5 = -4
(-4)2 = 16
3
3 - 5 = -2
(-2)2 = 4
4
4 - 5 = -1
(-1)2 = 1
4
4 - 5 = -1
(-1)2 = 1
4
4 - 5 = -1
(-1)2 = 1
5
5-5=0
(0)2 = 0
7
7-5=2
(2)2 = 4
8
8-5=3
(3)2 = 9
9
9-5=4
(4)2 = 16
Sum=?
“average” squared deviation = 52/(9-1) = 6.5
Standard deviation = square root of variance =
Describing Quantitative Data
3) Square each deviation.
(xi-mean)2
xi
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 Measuring
Sum=?
This is the variance.
6.5  2.55

We now have a choice between two descriptions for center
and spread

Mean and Standard Deviation

Median and Interquartile Range
Choosing Measures of Center and Spread
•The median and IQR are usually better than the mean and
standard deviation for describing a skewed distribution or a
distribution with outliers.
•Use mean and standard deviation only for reasonably
symmetric distributions that don’t have outliers.
•NOTE: Numerical summaries do not fully describe the
shape of a distribution. ALWAYS PLOT YOUR DATA!
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Choosing Measures of Center and Spread
Describing Quantitative Data

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Transformations
Think about this like a
curve on a test. If I add
Think about this like a
 Adding, subtracting, multiplying and dividing all the numbers in
5 points to everyone’s
a test.
If I add the data.
acurve
data seton
is called
“transforming”
test, how will the
5 points to everyone’s
 ONE OF THE CENTRAL CONCEPTS FROM THIS CHAPTER
spread of the grades
test, how will the class
is knowing how transformations affect measures of center and
change?
average
change?
spread.
I WILL
TEST ON THIS. OFTEN.
A LOT.
Mean
Standard deviation
Adding or subtracting
Mean gets added or
subtracted same
amount
Standard deviation
doesn’t change
Multiplying or dividing
Mean gets multiplied
or divided by same
amount
Standard deviation
gets multiplied or
divided by same
amount
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Example question

A researcher wishes to calculate the average height of patients
suffering from a particular disease. From patient records, the
mean was computed as 156 cm, and the standard deviation as
5 cm. Further investigation reveals that the scale was
misaligned, and that all readings are 2cm too large, for
example a patient whose height was really 180 cm was
measured as 182 cm. Furthermore the researcher would like
to work with statistics based on meters. The correct mean and
standard deviation are:
A) 1.56 m, 0.05 m
B) 1.54 m, 0.05 m
C) 1.56 m, 0.03 m
D) 1.58 m, 0.05 m
E) 1.58 m, 0.07 m
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