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Statistics
2.4: Measures of Variation
In this section we will learn different ways to measure the ___________________ of a data set.
________________ refers to how “spread out” the data values are from the ___________, usually the
____________ of the data set.
Objective 1: I can find the range of a data set.
The simplest measure of variation is the _____________.
R=
Read Example 1, pg 82
TIY 1: Find the range of the starting salaries for Corporation B. Compare your answer with Example 1
from the book.
Objective 2: I can find the deviation, variance, and standard deviation of a data set.
Symbols:
As a measure of variation, the range is very easy to compute. However, it only uses two values from the
data set, and it’s possible that one, or both of the values, are _______________. Two measures of
variation that use ______ the entries in a data set are the _________________ and the
____________________. But before we learn to calculate those 2 measures, we need to know about
___________________. The ______________ of an entry, ___ in a population data set is the
______________________________________________________________________.
Sample Mean:
Sample Deviation =
Population Mean:
Population Deviation =
TIY 2: Find the deviation of each starting salary for Corp B given in TIY 1. (Read “Insight”)
Salary: 40, 23, 41, 50, 49, 32, 41, 29, 52, 58
X
X-µ
Steps:
40
23
41
50
49
31
41
29
52
58
1. Find the mean:
2. Calculate each X - µ
3. Does ∑ (X - µ) = 0?
In Example 2, notice that the sum of the deviation is _______. Because this is true for any data set, it
doesn’t make sense to find the average of the deviations…it will also be 0, which is clearly not true. To
overcome this problem, we _________ the deviations to make them positive, and then find the
___________ of that sum. This is called the _________________ and it has a horrifying formula. The
______________________ of a population data set of _____ entries is:
*The symbol ____ is the lowercase Greek letter _________.
Variance = Symbol (______________) = Formula (
)
But since we have squared the deviation values (as well as their units), it does not make sense to use the
variance as a measure of variation, so we square root the final sum, which is called the
___________________________, as opposed to the “average deviation”. The ________________
____________________________ of a population data set of ____ entries is the square root of the
population _____________.
Its formula is:
Standard deviation = Symbol (_____________ ) = Formula (
)
Steps to finding variance and standard deviation by hand:
1. Find the _______ of the population data set. 1.
2. Find the ___________ of each entry.
2.
3. Square each __________.
3.
4. Add the deviations.
4.
5. Divide by _____ to get the _____________
_____________.
5.
6. Find the _______________ of the variance
to get the _____________________________.
6.
TIY 3: Find the population variance and SD of the starting salaries for Corp B.
2
Salary, x
Deviation , ( x   )
Squares, ( x   )
40
23
41
50
49
32
41
29
52
58
“On Average”, each data entry is about _____________________ units from the mean.
The formulas we used above for variance and Standard deviations were ____________________
formulas. If we have _____________ data, the formulas change slightly as do the variables we use to
represent them.
Sample Variance:
Sample Standard Deviation:
**Make sure you know what all of these symbols stand for when you come across them!!
TIY 4: Find the sample standard deviation of the starting salaries for Corp B.
NOTE: What is the major difference in the formula? ________________
Using technology to find the standard deviation – TI 84
Calculator Steps:
Sx: Sample Standard deviation
Ox: Population Standard deviation
TIY 5: Sample office rental rates for Seattle are listed below (in dollars/square foot). Use the graphing
calculator (or computer) to find the mean rental rate and the standard deviation.
40.00
43.00
46.00
40.50
35.75
39.75
32.75
36.75
35.75
38.75
38.75
36.75
38.75
39.00
29.00
35.00
42.75
32.75
40.75
35.25
Objective 3: I can interpret the standard deviation of a data set.
When interpreting the standard deviation, remember that it is a measure of the typical amount that an
entry deviates from the mean. The more the entries are spread out, the greater the standard deviation
will be for that data set.
Many real-life data sets have distributions that are approximately symmetrical and bell-shaped. Later in
the book (Chapter 5), we will study this kind of distribution in greater detail. For now, we are going to
look at 2 important rules that allow us to make statements about the population using the mean and
standard deviation of a data set.
Empirical Rule:
1.
2.
3.
Example 7: In a survey conducted by the National Center for Health Statistics, the sample mean height
of women in the US (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches.
Estimate the percent of women whose heights are between 64 inches and 69.42 inches.
TIY 7:
A) Use the mean and SD from example 7 to estimate the percents of heights that are between 61.29 and
72.13 inches.
B) Use the mean and SD from example 7 to estimate the percent of heights that are between 58.58
inches and 66.71 inches.
The Empirical rule applies only to distributions that are symmetrical. But what about other distributions
(uniform, skewed, none)? Then we use Chebychev’s Theorem.
1.
2.
TIY8: Apply Chebychev’s Theorem to Alaska using k = 3. Interpret the results.
Example 9: The weights of adult males from 20-59 is skewed-right. If the average weight is 192.6
pounds with a standard deviation of 3.8 pounds, what statements can you make about the weights of
males within:
(1) 2 standard deviations of the mean?
(2) 3 standard deviations of the mean?