Putting Statistics to Work 6 Measure of Variation

Why Variation Matters Consider the following waiting times for 11 customers at 2 banks. Big Bank (three lines): 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0 Best Bank (one line): 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8 Point out that although the average wait time is the same for both banks, the variation is decreased in the Best Bank scenario. Over time, the customer satisfaction will be higher since there are fewer surprises with long waits. Lead a discussion oriented around various local business establishments that use the two approaches. Which bank is likely to have more unhappy customers? → Big Bank, due to more surprise long waits

range = highest value (max) – lowest value (min) The range of a data set is the difference between its highest and lowest data values. range = highest value (max) – lowest value (min)

Example Consider the following two sets of quiz score for nine students. Which set has the greater range? Would you also say that the scores in the set are more varied? Quiz 1: 1 10 10 10 10 10 10 10 10 Quiz 2: 2 3 4 5 6 7 8 9 10

Example (cont) Solution The range for Quiz 1 is 10 – 1 = 9 points, which is greater than the range for Quiz 2 of 10 – 2 = 8 points. However, aside from a single low score (an outlier), Quiz 1 has no variation at all because every other student got a 10. In contrast, no two students got the same score on Quiz 2, and the scores are spread throughout the list of possible scores. The scores on Quiz 2 are more varied even though Quiz 1 has the greater range.

Quartiles The lower quartile (or first quartile) divides the lowest fourth of a data set from the upper three-fourths. It is the median of the data values in the lower half of a data set. The middle quartile (or second quartile) is the overall median. The upper quartile (or third quartile) divides the lower three-fourths of a data set from the upper fourth. It is the median of the data values in the upper half of a data set.

The Five-Number Summary The five-number summary for a data set consists of the following five numbers: low value lower quartile median upper quartile high value A boxplot shows the five-number summary visually, with a rectangular box enclosing the lower and upper quartiles, a line marking the median, and whiskers extending to the low and high values.

The Five-Number Summary Five-number summary of the waiting times at each bank: Big Bank Best Bank low value (min) = 4.1 lower quartile = 5.6 median = 7.2 upper quartile = 8.5 high value (max) = 11.0 low value (min) = 6.6 lower quartile = 6.7 median = 7.2 upper quartile = 7.7 high value (max) = 7.8 The corresponding boxplot: Demonstrate the importance of the five-number summary by showing how the corresponding box plots provide excellent visual feedback in terms of variability and measure of center. Have students find the summary and build box plots using contemporary data sets that give a strong sense of context.

Note: This is a sample standard deviation formula. The standard deviation is the single number most commonly used to describe variation. Note: This is a sample standard deviation formula.

Calculating the Standard Deviation The standard deviation is calculated by completing the following steps: 1. Compute the mean of the data set. Then find the deviation from the mean for every data value. deviation from the mean = data value – mean 2. Find the squares of all the deviations from the mean. 3. Add all the squares of the deviations from the mean. 4. Divide this sum by the total number of data values minus 1. 5. The standard deviation is the square root of this quotient.

The Range Rule of Thumb The standard deviation is approximately related to the range of a data set by the range rule of thumb: If we know the standard deviation for a data set, we estimate the low and high values as follows: The reason for the 4 is due to the idea that the usual high is approximately 2 standard deviations above the mean and the usual low is approximately 2 standard deviations below. Put this rule into action by asking the class what an approximate mean is for the height of an American male 18–24 years old. Many will guess around 5’10,” which is what the National Center for Health Statistics reports. Then ask for an approximate standard deviation. Most students are very unsure of this statistic. Show how the range rule can work as an estimating tool. Going in the reverse direction, consider using the average pregnancy length of 268 days with a standard deviation of 15 days. Ask students to estimate a usual low and a usual high pregnancy length.

Example Studies of the gas mileage of a Prius under varying driving conditions show that it gets a mean of 45 miles per gallon with a standard deviation of 4 miles per gallon. Estimate the minimum and maximum gas mileage that you can expect under ordinary driving conditions. Solution low value ≈ mean – (2 × standard deviation) = 45 – (2 × 4) = 37

Example (cont) high value ≈ mean + (2 × standard deviation) = 45 + (2 × 4) = 53 The range of gas mileage for the car is roughly from a minimum of 37 miles per gallon to a maximum of 53 miles per gallon.