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 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).

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.

Inter quartile Range

Quartiles: Inter quartile : IQR = Q3 – Q1

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.

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 80th percentile 80% included here

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.

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).

Variance: Computational Formula Population Sample

Variance Use the computational formula to calculate the variance.

Variability Example: Variance Las Vegas Hotel Rates

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.

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,

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

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

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

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.

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.

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, 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 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

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

ANY QUESTIONS

THANK YOU