STATISTICS ELEMENTARY MARIO F. TRIOLA Section 2-5 Measures of Variation MARIO F. TRIOLA EIGHTH EDITION
Compute measures of variability. Objective Compute measures of variability.
Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 6.7 7.3 7.7 7.4 7.7 7.7 8.5 7.7 9.3 7.7 10.0
Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 6.7 7.3 7.7 7.4 7.7 7.7 8.5 7.7 9.3 7.7 10.0 Jefferson Valley Bank Bank of Providence Mean Median Mode Midrange 7.15 7.20 7.7 7.10 7.15 7.20 7.7 7.10
Section 2.5 Measures of Variation This last example shows that each bank has the same measures of center, but a closer look at the distribution of waiting times shows that the variability of waiting times is not the same.
Dotplots of Waiting Times Even though the measures of center are all the same, it is obvious from the dotplots of each group of data that there are some differences in the ‘spread’ (or variation) of the data. page 69 of text Figure 2-14
Measures of Variation Range Standard Deviation Variance Interquartile Range
Range Measures of Variation value greatest least value The range is not a very useful measure of variation as it only uses two values of the data. Other measures of variation (to follow) will more more useful as they are computed by using every data value.
Range Jefferson Valley Providence Jefferson Valley Providence 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 7.3 7.7 7.4 8.5 9.3 10.0 Jefferson Valley Range = 7.7 - 6.5 = 1.2 minutes Providence Range = 10.0 – 4.2 = 5.8 minutes
Interquartile Range IQR = Q3 – Q1 Measures of Variation Interquartile Range IQR = Q3 – Q1 The range is not a very useful measure of variation as it only uses two values of the data. Other measures of variation (to follow) will more more useful as they are computed by using every data value.
Interquartile Range (IQR) = Q3-Q1 Jefferson Valley Providence 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 7.3 7.7 7.4 8.5 9.3 10.0 Jefferson Valley Median = 7.2 Q1 = 6.7 Q3 = 7.7 IQR = 7.7 - 6.7 = 1.0 Providence Q1 = 5.8 Q3 = 8.5 IQR = 8.5 – 5.8 = 2.7 minutes
Measures of Variation Standard Deviation a measure of variation of the scores about the mean (average deviation from the mean) Ask students to explain what this definition indicates.
Sample Standard Deviation Formula
Sample Standard Deviation Formula (x - x)2 S = n - 1 Formula 2-4 The definition indicates that one should find the average distance each score is from the mean. The use of n-1 in the denominator is necessary because there are only n-1 independent values - that is, only n-1 values can be assigned any number before the nth value is determined. page 70 of text
Computing Standard Deviation Compute the mean x
Computing Standard Deviation Compute the mean Subtract the mean from each data value x x-x
Computing Standard Deviation Compute the mean Subtract the mean from each data value Square the differences x x-x
Computing Standard Deviation Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences x x-x
Computing Standard Deviation Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences Divide the sum by (n-1) x x-x
Computing Standard Deviation Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences Divide the sum by (n-1) Take the square root of this result x x-x
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean Subtract the mean from each data value
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean Subtract the mean from each data value Square the differences
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences Divide the sum by (n-1)
Computing Standard Deviation Providence Bank #5 pg81 Compute the mean Subtract the mean from each data value Square the differences Sum the squared differences Divide the sum by (n-1) Take the square root of this result
Population Standard Deviation (x - µ) 2 = N is the lowercase Greek ‘sigma’. Note the division by N(population size), rather than n-1 Most data is a sample, rather than a population; therefore, this formula is not use very often.
for Standard Deviation Symbols for Standard Deviation Sample Population s Sx xn-1 Textbook x xn Book Some graphics calculators Some graphics calculators Some non-graphics calculators Some non-graphics calculators Articles in professional journals and reports often use SD for standard deviation and VAR for variance.
Measures of Variation Variance
standard deviation squared Measures of Variation Variance standard deviation squared page 74 of text
standard deviation squared Measures of Variation Variance standard deviation squared s } 2 use square key on calculator After computing the standard deviation, square the resulting number using a calculator. Notation 2
Variance (x - x )2 (x - µ)2 n - 1 2 = N s2 = Sample Variance The formulas for variance are the same as for standard deviation without the square root - remind student, squaring a square root will result in the radicand. (x - µ)2 N 2 = Population Variance
Round-off Rule for measures of variation Carry one more decimal place than is present in the original set of values. Round only the final answer, never in the middle of a calculation. Same rounding rule as for measures of center. page 75 of text
HOMEWORK Page 81 3, 5
Standard Deviation from a Frequency Table
Standard Deviation from a Frequency Table Rating Frequency 0-2 20 3-5 14 6-8 15 9-11 2 12-14 1 midpoints 1 4 7 10 13 20 56 105 13
Rating Frequency 0-2 20 3-5 14 6-8 15 9-11 2 12-14 1 midpoints 1 4 7 10 13 -3.1153846 -0.11538461 2.88461538 5.88461538 8.88461538
Standard Deviation from a Frequency Table Rating Frequency 0-2 20 3-5 14 6-8 15 9-11 2 12-14 1 midpoints 1 4 7 10 13 9.705621301 0.013313609 8.32100591 34.6286982 78.9363905
Standard Deviation from a Frequency Table Rating Frequency 0-2 20 3-5 14 6-8 15 9-11 2 12-14 1 midpoints 1 4 7 10 13 9.705621301 0.013313609 8.32100591 34.6286982 78.9363905 194.1124260 0.186390532 124.815088 69.2573964 78.9363905
Standard Deviation from a Frequency Table
HOMEWORK Page 81-84 9, 11, 29
Objective: Understanding Standard Deviation Apply the Empirical Rule Apply Chebyshev’s Rule Apply Range Rule of Thumb Identify Unusual Values
Estimation of Standard Deviation Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range 4s or Reminder: range is the highest score minus the lowest score
Estimation of Standard Deviation Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range 4s or Reminder: range is the highest score minus the lowest score Range 4 s
Estimation of Standard Deviation highest value - lowest value Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range 4s or Reminder: range is the highest score minus the lowest score Range 4 highest value - lowest value s = 4
Usual Sample Values
minimum ‘usual’ value (mean) - 2 (standard deviation) Usual Sample Values minimum ‘usual’ value (mean) - 2 (standard deviation) minimum x - 2(s)
Usual Sample Values minimum x - 2(s) maximum x + 2(s) minimum ‘usual’ value (mean) - 2 (standard deviation) minimum x - 2(s) maximum ‘usual’ value (mean) + 2 (standard deviation) maximum x + 2(s) These ideas will be used repeatedly throughout the course.
(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 page 79 of text x
(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 68% within 1 standard deviation Some student have difficulty understand the idea of ‘within one standard deviation of the mean’. Emphasize that this means the interval from one standard deviation below the mean to one standard deviation above the mean. 34% 34% x - s x x + s
(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 13.5% 13.5% x - 2s x - s x x + s x + 2s
The Empirical Rule x - 3s x - 2s x - s x x + s x + 2s x + 3s (applies to bell-shaped distributions) FIGURE 2-15 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution. 34% 34% 2.4% 2.4% 0.1% 0.1% 13.5% 13.5% x - 3s x - 2s x - s x x + s x + 2s x + 3s
Example Application of the Empirical Rule A set of 1000 test scores has a symmetric, mound-shaped distribution. The mean is 175 and the standard deviation is 10. Approximately what percent of the scores are between 175 and 195?
of the area within 2 standard deviations 95% of scores are between The shaded area is half of the area within 2 standard deviations of the mean .... so (.5)(.95)=.475 175 185 195
Example Application of the Empirical Rule A set of 1000 test scores has a symmetric, mound-shaped distribution. The mean is 175 and the standard deviation is 10. Approximately how many scores are between 155 and 165?
Example Application of the Empirical Rule The area from 155 to 195 is 0.95 The area from 165 to 185 is 0.68 Subtracting these values gives the area from 155 to 165 and 185 to 195 combined. We need to divide the result by 2 because the symmetry splits this area equally. 0.95 - 0.68 = 0.27 0.27 divided by 2 = 0.135 Example Application of the Empirical Rule 165 195 155 175 185
Chebyshev’s Theorem applies to distributions of any shape. the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K2 , where K is any positive number greater than 1. at least 3/4 (75%) of all values lie within 2 standard deviations of the mean. at least 8/9 (89%) of all values lie within 3 standard deviations of the mean. Emphasize Chebyshev’s Theorem applies to data that is in a distribution of any shape - that is, it is less prescriptive than the Empirical Rule. page 80 of text
Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations. This idea will be revisited throughout the study of Elementary Statistics.
Homework Page 81-84 19, 20, 21, 22, 23, 24, 25, 30 a and b