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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 _____________.
The range =
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.
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 Deviation =
Population Deviation =
Read Example 2, pg 83
TIY 2: Find the deviation of each starting salary for Corp B given in TIY 1. (Read “Insight”)
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 _________.
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
_________________________ (or just SD) of a population data set of ____ entries is the square
root of the population _____________.
Its formula is:
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.
Read Example 3, pg 84.
Read “Study Tip”, pg 84.
TIY 3: Find the population variance and SD of the starting salaries for Corp B.
Salary, x
Deviation , ( x   ) Squares, ( x   ) 2
The formulas we used above for variance and SD 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!!
Read Example 4, pg 85
TIY 4: Find the sample standard deviation of the starting salaries for Corp B.
Using technology to find the standard deviation
Read Example 5, pg 87
Calculator steps:
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?