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: Variance  s2 = Σ(x – x)2 n – 1 Standard Deviation  s = √ s2 *Note, when finding variance/standard deviation of a Sample, divide by (n – 1) instead of N

EX: find variance and standard deviation of the sample:

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.

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

Standard Deviation for grouped data: Variance  s2 = Σ(x – x)2 f n – 1 (Use class midpoint for x) Standard Deviation  s = √ s2

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

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

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