Section 2.4 Measures of Variation

Range The difference between the maximum and the minimum data entries in a data set. Range = max value – min value

Deviation The difference between a data entry (x) and the mean (µ) Deviation of x = x - µ

EX: find the range of the set and the deviation of each value Salary (1000 s of dollars) Deviation 41 37 39 45 47

Population Variance (σ2) Square the deviations of the data set, then average them to get the population variance. σ2 = Σ(x - µ)2 n

Population Standard Deviation … Just take the square root of the population variance. (Symbol = σ)

EX: find the variance and standard deviation of the data set 41 37 39 45 47

Sample variance and standard deviation:  

EX: find variance and standard deviation of the sample: The weights (in pounds) of a sample of 10 U.S. presidents. 173 175 200 173 160 185 195 230 190 180

Interpreting Standard Deviation Standard deviation is the measure of the typical amount an entry deviates from the mean. The more entries are spread out, the greater the standard deviation.

Empirical Rule For data with a symmetric (bell-shaped) distribution, the standard deviation has the following characteristics: 1. About 68% of the data lie within 1 standard deviation of the mean. 2. About 95% of the data lie within 2 standard deviations of the mean. 3. About 99.7% of the data lie within 3 standard deviations of the mean.

Ex (from page 96) Use the Empirical Rule – assume the data has a bell-shaped distribution: 30. The mean monthly utility bill for a sample of households in a city is $70, with a standard deviation of $8. Between what two values do about 95% of the data lie?

Chebychev’s Theorem k2 This works for ANY data set, symmetric or not. The portion of any data set lying within k standard deviations of the mean is at least 1 - 1 k2

Ex: (from page 96) 36. Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 32, the mean duration of Old Faithful’s eruptions is 3.32 minutes and the standard deviation is 1.09 minutes. Using Chebyshev’s Theorem, determine at least how many of the eruptions lasted between 1.14 minutes and 5.5 minutes.

Standard Deviation for grouped data:  

Section 2.5 Measures of Position

Quartiles Data set is divided into 4 sections, separated by 3 QUARTILES Q1 – about 25% of the data is below Quartile 1 Q2 – about 50% of the data is below Quartile 2 Q3 – about 75% of the data is below Quartile 3 (Q2 is also the median!)

Ex: Find the Quartiles The number of vacations days used by a sample of 20 employees in a recent year: 3 9 2 1 7 5 3 2 2 6 4 0 10 0 3 5 7 8 6 5

InterQuartile Range (IQR) IQR is the measure of variation that given the range of the middle 50% of the data. It is the difference between the 3rd and 1st quartiles. IQR = Q3 – Q1

Box-and-Whisker Plot Find the 3 quartiles of the data set, and the minimum and maximum entries Construct a horizontal scale that spans the range. Draw a box from Q1 to Q3 and draw a vertical line at Q2. Draw whiskers from the box to the min and max entries.

Construct a box-and-whisker plot The number of vacations days used by a sample of 20 employees in a recent year: 3 9 2 1 7 5 3 2 2 6 4 0 10 0 3 5 7 8 6 5

Percentiles and Deciles Similar to Quartiles, but the data is divided into 10 or 100 parts instead of 4. 8th Decile  80% of the data falls before the decile. 95th Percentile  95% of the data falls before the percentile

Standard Score (z-score) Represents the number of standard deviations a given value (x) falls from the mean (µ). z = value – mean = x - µ standard deviation σ

Ex: (from page 112) 48. The life spans of a species of fruit fly have a bell-shaped distribution, with a mean of 33 days and a standard deviation of 4 days. A. The life spans of three randomly selected fruit flies are 34 days, 30 days, and 42 days. Find the z-score that corresponds to each life span. Determine if any of these life spans are unusual. B. The life spans of three randomly selected fruit flies are 29 days, 41 days, and 25 days. Using the Empirical Rule, find the percentile that corresponds to each life span.