Lecture 4: Measures of Variation

1
Lecture 4 Measures of Variation
Review of Lecture 3 Measures of Center

• Given a stem and-leaf plot
• Be able to find
• Mean
• (4042350512526467)/1046.7
• Median
• (5051)/250.5
• mode
• 50

5th 6th

• Given a regular frequency distribution
• Be able to find
• Sample size
• 245161340
• Mean
• (81210160)/401.15
• Median
• average of the two middle values1

Median group
2
2.5 Measures of Variation
Measure of Variation (Measure of Dispersion)
A measure helps us to know the spread
of a data set.
Candidates Range Standard
Deviation Variance
Coefficient of Variation
Statistics handles variation. Thus this section
one of the most important sections in the entire
book
3
Definition

• The range of a set of data is the difference
  between the highest value and the lowest value

Range(Highest value) (Lowest value)
Example Range of 1 3 14 is 14-113.
4
Standard Deviation
The standard deviation of a set of values is a
measure of variation of values about the mean

• We introduce two standard deviation
• Sample standard deviation
• Population standard deviation

5
Sample Standard Deviation Formula
Data value
Sample size
6
Sample Standard Deviation (Shortcut Formula)
7
Example Publix check-out waiting times in minutes

• Data 1 4 10. Find the sample mean and sample
  standard deviation.

Using the shortcut formula
1-5
n3
8
Standard Deviation - Key Points

• The standard deviation is a measure of variation
  of all values from the mean

• The value of the standard deviation s is usually
  positive and always non-negative.

• The value of the standard deviation s can
  increase dramatically with the inclusion of one
  or more outliers (data values far away from all
  others)

• The units of the standard deviation s are the
  same as the units of the original data values

9
Population Standard Deviation
This formula is similar to Formula 2-4 but
instead the population mean and population size
are used
10
Variance

• The variance of a set of values is a measure of
  variation equal to the square of the standard
  deviation.

• Sample variance s2 Square of the sample
  standard deviation s

11
Variance - Notation
standard deviation squared
12
Round-off Rulefor Measures of Variation

• Carry one more decimal place than is present in
  the original set of data.

Round only the final answer not values in the
middle of a calculation.
13
Definition
The coefficient of variation (or CV) for a set of
sample or population data expressed as a
percent describes the standard deviation
relative to the mean

• A measure good at comparing variation between
  populations
• No unit makes comparing apple and pear possible.

14
Example How to compare the variability in
heights and weights of men

• Sample 40 males were randomly selected. The
  summarized statistics are given below.

Conclusion Heights (with CV4.42) have
considerably less variation than weights (with
CV15.26)
Solution Use CV to compare the
variability Heights Weights
15
Standard Deviation from a Frequency Distribution

• Use the class midpoints as the x values

16
Example Number of TV sets Owned by households

• A random sample of 80 households was selected
• Number of TV owned is collected given below.

• Compute
• the sample mean
• (b) the sample standard deviation

17
Estimation of Standard Deviation Range Rule of
Thumb
For estimating a value of the standard deviation
s Use Where range (highest value) (lowest
value)
18
Estimation of Standard Deviation Range Rule of
Thumb
For interpreting a known value of the standard
deviation s find rough estimates of the minimum
and maximum usual values by using
19
Definition
Empirical (68-95-99.7) Rule For data sets having
a distribution that is approximately bell shaped
the following properties apply

• About 68 of all values fall within 1 standard
  deviation of the mean

• About 95 of all values fall within 2 standard
  deviations of the mean

• About 99.7 of all values fall within 3 standard
  deviations of the mean

20
The Empirical Rule
FIGURE 2-13
21
The Empirical Rule
FIGURE 2-13
22
The Empirical Rule
FIGURE 2-13
23
Recap
In this section we have looked at

• Range

• Standard deviation of a sample and population

• Variance of a sample and population

• Coefficient of Variation (CV)

• Standard deviation using a frequency distribution

• Range Rule of Thumb

• Empirical Distribution

24
Homework Assignment 4

• problems 2.5 1 3 7 9 11 17 23 25 27 31
• Read section 2.6 Measures of relative standing.