E Spring Chapter 3 Summary Statistics

2 Measures of Central Location/Central Tendency Mean, Median, Mode Measures of Variability/Dispersion Range, Standard Deviation, Variance, Coefficient of Variation Measures of Shape Skewness (e.g. Pearson’s 2 nd Skewness) Summarizing - Distribution

Three Measures of Central Tendency Tendency 3 StatisticFormula Excel Formula ProCon Mean=AVERAGE(Data) Familiar and uses all the sample information. Influenced by extreme values. Median Middle value in sorted array =MEDIAN(Data) Robust when extreme data values exist. Ignores extremes and can be affected by gaps in data values.

Three Measures of Central Tendency (cont.) 4

Variance 5 Note! the denominator is sample size (n) minus one !

Variance(cont’d) 6  Excel’s built in functions are Statistic Excel population formula Excel sample formula Variance=VAR.P(Array)=VAR.S(Array) Standard deviation =STDEV.P(Array)=STDEV.S(Array)

Coefficient of Variation 7  The coefficient of variation(CV) of a set of observations is the standard deviation of the observations divided by their mean, that is: This coefficient provides a unit-free measure of variation.  It measures relative dispersion, and is useful for comparing dispersion of variables measured in different units or with different means.

Measure of Skewness 8  Pearson’s Skewness Coefficients First: S k = (mean-mode)/sample averg(or pop averg) Second: S k = 3(mean-median)/sample averg(or pop averg)  Characteristics of Pearson’s Second Skewness Coefficient: Usually exist between -3 and +3 zero means symmetric. Negative means negative (left) skewness Positive means positive (right) skewness

Mean, Median, Mode If a distribution is right-skewed (positive) it is often true: MEAN > MEDIAN > MODE If a distribution is left-skewed (negative) it is often true: MODE > MEDIAN > MEAN

Excel =AVERAGE(Array): Returns the arithmetic mean. =MEDIAN(Array): Returns the median of an ordered array. The array must be put in order before use, or the value it returns is meaningless. =MODE.SNGL(Array): Returns the first mode that is found in an array. =MODE.MULT(Array): Will return multiple modes if they exist in an array. =MIN(Array): Returns the value of the smallest magnitude in an array. =MAX(Array): Returns the value of the greatest magnitude in an array. =VAR.P(Array): Returns the population variance of an array. =VAR.S(Array): Returns the sample variance of an array. =STDEV.P(Array): Returns the population standard deviation of an array. =STDEV.S(Array): Returns the sample standard deviation of an array. Data==>Data Analysis==>Descriptive Statistics: Generates a table of statistics for one or more variables.