Putting Statistics to Work: Measure of Variation and Why Variation Matters

1. Putting Statistics to Work 6 Measure of Variation

2. 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 Which bank is likely to have more unhappy customers? → Big Bank, due to more surprise long waits

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

4. 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

5. 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.

6. 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.

7. 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.

8. 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:

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

10. 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.

11. 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:

12. 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

13. 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.