Measures of Dispersion or Measures of Variability
Measures of Variability • A single summary figure that describes the spread of observations within a distribution.
MEASURES OF DESPERSION • RANGE • INTERQUARTILE RANGE • VARIANCE • STANDARD DEVIATION
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.
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: 891-52 = 839
Pros 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). Pros and Cons of the Range
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.
Quartiles: Inter quartile : IQR = Q3 – Q1
Pros Fairly easy to compute. Scores exist in the data set. Eliminates influence of extreme scores. Cons Discards much of the data. Pros and Cons of the Interquartile Range
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)
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
Locating Percentiles in a Frequency Distribution 25% included here 25th percentile 50% included here 50th percentile 80thpercentile 80% included here
Five Number Summary • Minimum Value • 1st Quartile • Median • 3rd Quartile • Maximum Value
VARIANCE: Deviations of each observation from the mean, then averaging the sum of squares of these deviations. STANDARD DEVIATION: “ ROOT- MEANS-SQUARE-DEVIATIONS”
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).
Population Sample Variance: Computational Formula
Variance • Use the computational formula to calculate the variance.
Variability Example: Variance • Las Vegas Hotel Rates
Pros 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. Pros and Cons of Variance
Standard Deviation • To “undo” the squaring of difference scores, take the square root of the variance. • Return to original units rather than squared units.
Quantifying Uncertainty • Standard deviation: measures the variation of a variable in the sample. • Technically,
Population Sample Standard Deviation Measure of the average amount by which observations deviate on either side of the mean. The square root of the variance.
Variability Example: Standard Deviation Mean: 6 Standard Deviation: 2
Variability Example: Standard Deviation Mean: $371.60 Standard Deviation:
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. Cons Influenced by extreme scores. Pros and Cons of Standard Deviation
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.
A 100 samples were selected. Each of the sample contained 100 normal individuals. The mean Systolic BP of each sample is presented 110, 120, 130, 90, 100, 140, 160, 100, 120, 120, 110, 130, etc Mean = 120 Sd., = 10 Systolic BP level No. of samples 90 - 5 100 - 10 110 - 20 120 - 34 130 - 20 140 - 10 150 - 5 160 - 2
Normal Distribution Mean = 120 SD = 10 120 100 90 110 130 140 150 • 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
WHICH MEASURE TO USE ? DISTRIBUTION OF DATA IS SYMMETRIC ---- USE MEAN & S.D., DISTRIBUTION OF DATA IS SKEWED ---- USE MEDIAN & QUARTILES