1. MEASURES OF SPREAD – VARIABILITY- DIVERSITY- VARIATION-DISPERSION
Variability in Nature, life, and various processes we
investigate is fundamental to the theory of Statistics.

2. Measures of variability
Range, Inter-Quartile Range (IQR), Standard Deviation
RANGE is the difference between the largest and the smallest observations.
Range = maximum value – minimum value
The more variability or spread is in the data, the larger the difference between the min and max, the larger the range.
Example. For the two data sets summarized in the histograms:
A: min = 6.96, max = 13, range= = 6.04
B: min = , max = , range= (-22.74)=63.67

3. INTER-QUARTILE RANGE (IQR)
Percentiles – divide data into 100ths.
E.g. GRE score in 85th percentile means that 85 percent of the students taking the test scored lower and 15 percent of the students scored higher than this.
QUARTILES- special percentiles: 50th, 25th and 75th percentiles.
50th percentile= median.
25th percentile = the first quartile, Q1= median of the lower half of the data.
75th percentile = the third quartile, Q3 = median of the upper half of the data.
Example. Data is number of km to school for a sample of 18 kids.
Sorted data:
Q median Q3
Median=(4+5)/2=4.5
Data below median: ; median of this data: 3 = Q1
Data above the median: ; median of this data: 7 = Q3.
Inter-Quartile Range, IQR= Q3- Q1=7-3=4.

4. FIVE POINT SUMMARY: (MIN, Q1, MEDIAN, Q3, MAX)
Example. Five point summary of the distance to school data.
Min. Q1 Median Q3 Max.
BOXPLOTS – Graphical display of the five point summary
Example. Boxplot of the distance to school data.
Box: between Q1 and Q3.
Line inside the box-median.
Whiskers drawn to max/min.

5. Boxplot with outliers Whiskers drawn to if no outliers to max/min.
if high/low outliers present to 1.5xIQR above/below Q3/Q1;
Outlier(s) indicated as an open circle or a line above/below the whiskers.
Example. Boxplot of the distance to school data with an outlier.

6. STANDARD DEVIATION
STANDARD DEVIATION – measures average deviation of the data from the mean.
= deviation of the first observation from the sample mean
= deviation of the ith observation from the sample mean
Could try as the average deviation from the mean.
Problem: Difficult mathematically to deal with because of the absolute value.
Solution. Use squared deviations.

7. SAMPLE VARIANCE
SAMPLE VARIANCE = =

8. SAMPLE VARIANCE AND STANDARD DEVIATION
Notes on sample variance
Sample variance is always nonnegative,
Sample variance = 0 only if all deviations from the mean are zero, that is all observations are the same.
SAMPLE STANDARD DEVIATION S =POSITIVE SQUARE ROOT OF SAMPLE VARIANCE S=
Example. Standard deviations of the distance to school data with and without an outlier.
Original data set: variance: = 7.87, standard deviation= 2.81.
Data set with an outlier: variance = 40.57, standard deviation= 6.37.
UNITS: All measures of center are in the same units as the observations.
Variance is in the squared units of the observations.
Standard deviation is in the same units as the observations.

9. MEASURES OF CENTER AND SYMMETRY OF THE DATA
Median
Mean
Symmetric histogram
Median
Mean
Skewed to the right
Median ?
Mean ?
Skewed to the left

10. MEASURES OF CENTER AND SYMMETRY OF THE DATA - SUMMARY
HISTOGRAM SKEWED to the RIGHT
MEAN > MEDIAN
HISTOGRAM SYMMETRIC
MEAN=MEDIAN
HISTOGRAM SKEWED to the LEFT
MEAN < MEDIAN

11. HOW CLOSE TO THE MEAN IS THE MAJORITY OF DATA??
Interpretation of S
HOW CLOSE TO THE MEAN IS THE MAJORITY OF DATA??
Chebyshev’s inequality (any data)
(mean - ks, mean + ks) contains AT LEAST
(1-1/k*k)100% of the data
k=1: 0% k=2: 75% k=3: 89%
Empirical rule (symmetric, bell-shaped)
k=1: 68% k=2: 95% k=3: 99.7%