1. Measures of Dispersion or Measures of Variability

2. Measures of Variability
A single summary figure that describes the spread of observations within a distribution.

3. MEASURES OF DESPERSION
RANGE
INTERQUARTILE RANGE
VARIANCE
STANDARD DEVIATION

4. Measures of Variability
Range
Difference between the smallest and largest observations.
Interquartile Range
Range of the middle half of scores.
Variance
Mean of all squared deviations from the mean.
Standard Deviation
Measure of the average amount by which observations deviate from the mean. The square root of the variance.

5. Variability Example: Range
Las Vegas Hotel Rates
52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891
Range: = 839

6. Pros and Cons of the Range
Very easy to compute.
Scores exist in the data set.
Cons
Value depends only on two scores.
Very sensitive to outliers.
Influenced by sample size (the larger the sample, the larger the range).

7. Inter quartile Range The inter quartile range is Q3-Q1
50% of the observations in the distribution are in the inter quartile range.
The following figure shows the interaction between the quartiles, the median and the inter quartile range.

8. Inter quartile Range

9. Quartiles:
Inter quartile :
IQR = Q3 – Q1

13. Pros and Cons of the Interquartile Range
Fairly easy to compute.
Scores exist in the data set.
Eliminates influence of extreme scores.
Cons
Discards much of the data.

14. Percentiles and Quartiles
Maximum is 100th percentile: 100% of values lie at or below the maximum
Median is 50th percentile: 50% of values lie at or below the median
Any percentile can be calculated. But the most common are 25th (1st Quartile) and 75th (3rd Quartile)

15. Locating Percentiles in a Frequency Distribution
A percentile is a score below which a specific percentage of the distribution falls.
The 75th percentile is a score below which 75% of the cases fall.
The median is the 50th percentile: 50% of the cases fall below it
Another type of percentile :The lower quartile is 25th percentile and the upper quartile is the 75th percentile

16. Locating Percentiles in a Frequency Distribution
25% included here
25th percentile
50% included here
50th percentile
80th percentile
80% included
here

17. Five Number Summary Minimum Value 1st Quartile Median 3rd Quartile
Maximum Value
These five measures give a good description of any distribution that is relatively bump shaped (has only one mode), regardless of whether it is symmetrical or not.
However, if the distribution is relatively symmetrical, there are other measures we can use that will help us more than these measures.

18. VARIANCE:
Deviations of each observation from the mean, then averaging the sum of squares of these deviations.
STANDARD DEVIATION:
“ ROOT- MEANS-SQUARE-DEVIATIONS”

19. Variance
The average amount that a score deviates from the typical score.
Score – Mean = Difference Score
Average of Difference Scores = 0
In order to make this number not 0, square the difference scores ( negatives to become positives).

20. Variance: Computational Formula
Population
Sample

21. Variance
Use the computational formula to calculate the variance.

22. Variability Example: Variance
Las Vegas Hotel Rates

23. Pros and Cons of Variance
Takes all data into account.
Lends itself to computation of other stable measures (and is a prerequisite for many of them).
Cons
Hard to interpret.
Can be influenced by extreme scores.

24. Standard Deviation
To “undo” the squaring of difference scores, take the square root of the variance.
Return to original units rather than squared units.

25. Quantifying Uncertainty
Standard deviation: measures the variation of a variable in the sample.
Technically,

26. Standard Deviation Population Sample
Measure of the average amount by which observations deviate on either side of the mean. The square root of the variance.
Population
Sample

27. Variability Example: Standard Deviation
Mean: 6
Standard Deviation: 2

28. Variability Example: Standard Deviation
Mean: $371.60
Standard Deviation:

33. Pros and Cons of Standard Deviation
Influenced by extreme scores.
Pros
Lends itself to computation of other stable measures (and is a prerequisite for many of them).
Average of deviations around the mean.
Majority of data within one standard deviation above or below the mean.

34. Mean and Standard Deviation
Using the mean and standard deviation together:
Is an efficient way to describe a distribution with just two numbers.
Allows a direct comparison between distributions that are on different scales.

35. Systolic BP level No. of samples 90 - 5 100 - 10 110 - 20 120 - 34
A 100 samples were selected. Each of the sample contained 100 normal individuals. The mean Systolic BP of each sample is presented
110, , 130, , ,
140, , , , , 110, , etc
Mean = 120
Sd., = 10
Systolic BP level No. of samples

36. Normal Distribution Mean = 120 SD = 10 90 100 120 110 130 140 150
The curve describes probability of getting any range of values ie., P(x>120), P(x<100), P(110 <X<130)
Area under the curve = probability
Area under the whole curve =1
Probability of getting specific number =0, eg P(X=120) =0

38. WHICH MEASURE TO USE ? DISTRIBUTION OF DATA IS SYMMETRIC
---- USE MEAN & S.D.,
DISTRIBUTION OF DATA IS SKEWED
---- USE MEDIAN & QUARTILES

39. ANY QUESTIONS

40. THANK YOU