Summary Measures Central Tendency Mean Median Mode Midrange Quartile Midhinge Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

Measures of Central Tendency Central Tendency MeanMedianMode Midrange Midhinge

The Mean (Arithmetic Average) It is the Arithmetic Average of data values: The Most Common Measure of Central Tendency Affected by Extreme Values (Outliers) Mean = 5Mean = 6 Sample Mean

The Median Median = 5 Important Measure of Central Tendency In an ordered array, the median is the middle number. If n is odd, the median is the middle number. If n is even, the median is the average of the 2 middle numbers. Not Affected by Extreme Values

The Mode Mode = 9 A Measure of Central Tendency Value that Occurs Most Often Not Affected by Extreme Values There May Not be a Mode There May be Several Modes Used for Either Numerical or Categorical Data No Mode

Midrange A Measure of Central Tendency Average of Smallest and Largest Observation: Affected by Extreme Value Midrange Midrange = 5

Quartiles Not a Measure of Central Tendency Split Ordered Data into 4 Quarters Position of i-th Quartile: position of point 25% Q1Q1 Q2Q2 Q3Q3 Q i(n+1) i 4 Data in Ordered Array: Position of Q 1 = 2.50 Q1Q1 =12.5 = 1(9 + 1) 4

Midhinge A Measure of Central Tendency The Middle point of 1st and 3rd Quarters Not Affected by Extreme Values Midhinge = Data in Ordered Array: Midhinge =

Measure of Variation Difference Between Largest & Smallest Observations: Range = Ignores How Data Are Distributed: The Range Range = = Range = = 5

Measure of Variation Also Known as Midspread: Spread in the Middle 50% Difference Between Third & First Quartiles: Interquartile Range = Not Affected by Extreme Values Interquartile Range Data in Ordered Array: = = 5

Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Variance For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

Comparing Standard Deviations s = = = Value for the Standard Deviation is larger for data considered as a Sample. Data : N= 8 Mean =16

Comparing Standard Deviations Mean = 15.5 s = Data B Data A Mean = 15.5 s = Mean = 15.5 s = 4.57 Data C

Coefficient of Variation Measure of Relative Variation Always a % Shows Variation Relative to Mean Used to Compare 2 or More Groups Formula ( for Sample):

Comparing Coefficient of Variation Stock A: Average Price last year = $50 Standard Deviation = $5 Stock B: Average Price last year = $100 Standard Deviation = $5 Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5%

Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Right-Skewed Left-SkewedSymmetric Mean =Median =Mode Mean Median Mode Median Mean Mod e

Box-and-Whisker Plot Graphical Display of Data Using 5-Number Summary Median Q 3 Q 1 X largest X smallest

Distribution Shape & Box-and-Whisker Plots Right-SkewedLeft-SkewedSymmetric Q 1 Median Q 3 Q 1 Q 3 Q 1 Q 3