1. 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

2. A. Mean – Arithmetic Mean ( Average )
Ex:
B. Median – Midpoint of the Data – as many observations above as below

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

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

5. We can Transform Data to Change Distribution Shape

6. Mammal:Brain vs Body

7. Log(Brain) vs Log(Body)

8. Variability or Dispersion of Data
EX1: EX2: EX3:
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

9. 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 =

10. 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)

11. C. Variance and Standard Deviation
Ex: Xi
Deviation from Mean
Average Deviation =
Mean Absolute Deviation (MAD) =
Squared Deviations
Average Squared Deviation =
(Variance)

12. Sample Variance -
Sample Std Deviation -
Ex:
Ex:

13. 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 = (1/4) = 75% of obs will lie within 2 std dev of mean
K = (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 % Obs

14. Ex:1
+ 2•s =
- 2•s =
Ex:2
+ 2•s =
- 2•s =
Ex:3
+ 2•s =
- 2•s =

15. Shortcut Formula for the Variance
Shortcut/Machine Formula

16. Ex:
Xi2

18. Estimate Mean and Variance for Grouped Data
fj – Class Freq mj – Class Mark
Mean
Variance
Example:
Sales Freq 21

19. Example:
Age Freq fj*mj fj*mj2
17 –
21 –
25 –
29 – 40
Estimate Median:

20. Anscombe Quartet