Slide Slide 1 Section 3-3 Measures of Variation

Slide Slide 2 Key Concept Because this section introduces the concept of variation, which is something so important in statistics, this is one of the most important sections in the entire book. Place a high priority on how to interpret values of standard deviation.

Slide Slide 3 Bank Example, p. 93 1) Left side of the class: Find mean, median, mode, and midrange for the single line system Right side of the class: Find mean, median, mode, and midrange for the multi-line system 2) Put the result on the board. Do both systems have the same measure of center? 3) Examine/compare the two data sets. What is fundamentally different about them? Single Line Multiple lines

Slide Slide 4 Definition The range of a set of data is the difference between the maximum value and the minimum value. Range = (maximum value) – (minimum value) Bank 1: Variable waiting lines 666 Bank 2: Single waiting lines477 Bank 3: Multiple waiting lines1314

Slide Slide 5 Definition The standard deviation of a set of sample values is a measure of variation of values about the mean. If the values are close together: small s If the values are far apart: large s.

Slide Slide 6 Sample Standard Deviation Formula (3-4)  ( x - x ) 2 n - 1 s =s =

Slide Slide 7 Sample Standard Deviation (Shortcut Formula: 3-5) n ( n - 1) s =s = n  x 2 ) - (  x ) 2

Slide Slide 8 Banking Example Jefferson Valley (single line): Providence (multiple lines):

Slide Slide 9 Using the formula for standard deviation (TI83) 1)Put stuff in L1, L2. Do 2 variable stats. The mean for L1 is ____. Find the deviation of each value from the mean. (L3: L1- ___). 2)Show that the sum of deviations from Step 1 is 0. Will it always be 0? (sum L3) 3)We want to avoid the canceling out of the positive and negative deviations – so we square the deviations (L4: L3^2) 4)We need a mean of those squared deviations, so we find the mean by dividing by n-1 (degrees of freedom). (sum L4/(n-1)) 5)Track the units. If the original times are in minutes, the deviations are in minutes, the squared deviations are in minutes squared (what?), and the mean is in minutes squared (huh?) 6)Since minutes squared doesn’t make much sense, take the square root to get back to the original units. (sqr rt of ans).

Slide Slide 10 Example 1)Use the data set (1, 3, 14) from the single line system to find s using formula )YOU: Use the data set (4, 7, 7 minutes) from a multiple line system to find s using formula )Which standard deviation is smaller? So which line is better?

Slide Slide 11 Standard Deviation - Important Properties  The standard deviation is a measure of variation of all values from the mean.  The value of the standard deviation s is usually positive.  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.

Slide Slide 12 Population Standard Deviation 2  ( x - µ ) N  = This formula is similar to the previous formula, but instead, the population mean and population size are used.

Slide Slide 13  Population variance: Square of the population standard deviation Definition  The variance of a set of values is a measure of variation equal to the square of the standard deviation. (A general description of the amount that values vary among themselves) Also: dispersion/spread  Sample variance: Square of the sample standard deviation s

Slide Slide 14 Variance - Notation standard deviation squared s  2 2 } Notation Sample variance Population variance

Slide Slide 15 Round-off Rule for 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.

Slide Slide 16 Day 2 Warm Up: Heart Rate Activity Is there a difference between male and female heart rates? Male: Female: Compare the range, standard deviation, and variances of these samples.

Slide Slide 17 Estimation of Standard Deviation Range Rule of Thumb For estimating a value of the standard deviation s, Use Where range = (maximum value) – (minimum value) CRUDE estimate Based on the principal: For many data sets, the vast majority (95%) lie within 2 std. dev.’s of the mean Simple rule to help us interpret std. devs. Range 4 s s 

Slide Slide 18 Age of Best Actresses Use the range rule of thumb to find a rough estimate of the standard deviation of the sample of 76 ages of actresses who won Oscars. Max age: 80 Min age: 21

Slide Slide 19 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” sample values by using: Minimum “usual” value (mean) – 2 X (standard deviation) = Maximum “usual” value (mean) + 2 X (standard deviation) =

Slide Slide 20 Example A statistics professor finds the times (in seconds) required to complete a quiz have a mean of 180 sec and a standard deviation of 30 secs. Is a time of 90 secs unusual? Why or why not? YOU: Typical IQ tests have a mean of 100 and a standard deviation of 15. Use the range rule of thumb to find the usual IQ scores. Is a value of 140 unusual?

Slide Slide 21 Definition Empirical ( ) 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.

Slide Slide 22 The Empirical Rule

Slide Slide 23 The Empirical Rule (applies only to bell-shaped distributions)

Slide Slide 24 The Empirical Rule

Slide Slide 25 Example: IQ Scores IQ scores are bell shaped with a mean of 100 and a standard deviation of 15. What percent of IQ scores fall between 70 and 130?

Slide Slide 26 Definition Applies to any distribution, but results are approximate. Chebyshev’s Theorem The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1-1/K 2, where K is any positive number greater than 1.  For K = 2, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean.  For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.

Slide Slide 27 Example IQ scores have a mean of 100 and a standard deviation of 15 (pretend we don’t know that IQ scores are bell-shaped). According to Chebyshev’s Theorem, what can we conclude about: 1)75% of the IQ scores? 2)89% of the IQ scores?

Slide Slide 28 Rationale for using n-1 versus n The end of Section 3-3 has a detailed explanation of why n – 1 rather than n is used. The student should study it carefully.

Slide Slide 29 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.  Free of specific units of measure  For comparing variation for values taken from different populations Sample Population s x CV =  100%  CV =   100%

Slide Slide 30 Example: Statdisk Height and Weight data for the 40 males Data Set 1, Appendix B Although the difference in units makes it impossible to compare the two standard deviations (inches vs. pounds), we can compare CV’s (which have no units). Find the CV for weights and the CV for heights.