1 Lecture 4 Measures of Variation Review of Lecture 3 Measures of Center
Given a stem and-leaf plot
Be able to find
Given a regular frequency distribution
Be able to find
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.
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
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
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.
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