Descriptive Measures Descriptive Measure – A Unique Measure of a Data Set Central Tendency of Data Mean Median Mode 2) Dispersion or Spread of Data A. Range B. Quartiles & Percentiles C. Variance & Std Deviation

A. Mean – Arithmetic Mean ( Average ) Ex: 1 3 3 5 8 B. Median – Midpoint of the Data – as many observations above as below 1 3 3 5 8 1 3 3 5 8 10

C. Mode – Most Frequent Observation 1 3 3 5 8 Relationship between Mean, Median, & Mode 1) Symmetrical Distribution

2) Right Skewed Distribution (Positive Skew) 3) Left Skewed Distribution (Negative Skew)

We can Transform Data to Change Distribution Shape

Mammal:Brain vs Body

Log(Brain) vs Log(Body)

Variability or Dispersion of Data EX1: 2 3 3 4 EX2: 1 2 4 5 EX3: 0 2 4 6 A. Range = Maximum Obs – Minimum Obs Quartiles – Divide the Data into Four Equal Groups 25% Obs ≤ Q1 ≤ 75% Obs Lower Quartile 50% Obs ≤ Q2 ≤ 50% Obs Middle Quartile 75% Obs ≤ Q3 ≤ 25% Obs Upper Quartile

Interquartile Range – IQR = Q3 – Q1 Percentiles – the Pth Percentile is the Value such that at most P% of the Observations are Less and at most (100 – P)% of the Observations are Greater than the Value. Method: Multiply P*n: If result is integer, the Percentile is midpoint between this obs & next. If result is decimal, the Percentile is the next observation Q1 = Q2 = Q3 = P80 = P90 = P95 =

Q1 Q3 Q2 Minimum Maximum Box & Whisker Plot for Data: Distance of Obs from Box > 1.5 * IQR – Mild Outlier (*) Distance of Obs from Box > 3.0 * IQR – Extreme Outlier (0)

C. Variance and Standard Deviation Ex: Xi 1 3 3 5 8 Deviation from Mean Average Deviation = Mean Absolute Deviation (MAD) = Squared Deviations Average Squared Deviation = (Variance)

Sample Variance - Sample Std Deviation - Ex:2 1 3 3 5 6 6 Ex:3 1 3 5 7 14

Significance of the Standard Deviation Tchebysheff’s Theorem – (k > 1) At least (1-(1/k2)) of observations will lie within k std dev of the mean. K = 2 1-(1/4) = 75% of obs will lie within 2 std dev of mean K = 3 1-(1/9) = 89% of obs will lie within 3 std dev of mean Empirical Rule: For Normal Data µ ± 1σ 68% Obs µ ± 2σ 95% Obs µ ± 3σ 99.7% Obs

Ex:1 1 3 3 5 8 + 2•s = - 2•s = Ex:2 1 3 3 5 6 6 + 2•s = - 2•s = Ex:3 1 3 5 7 14 + 2•s = - 2•s =

Shortcut Formula for the Variance Shortcut/Machine Formula

Ex:4 1 3 3 5 8 9 11 Xi2

Estimate Mean and Variance for Grouped Data fj – Class Freq mj – Class Mark Mean Variance Example: Sales Freq 0 10 1 5 2 3 3 2 4 1 21

Example: Age Freq fj*mj fj*mj2 17 – 20 18 21 – 24 12 25 – 28 8 29 – 32 2 40 Estimate Median:

Anscombe Quartet