29 Mar 2012

School

Department

Course

Professor

Variance ợ₂

-a measure of dispersion of a set of data around the mean

-variance equals the average of the sum of all the squared deviations of the sample

Standard deviation ợ

-deviation of the distance from any single measure of a set to the mean of that set

Calculating Deviation

-compare the deviation from the mean for every measurement in the sample

-square each deviation

-sum the squares of the deviations [SS]

-average the squared deviations by dividing the sum by the number of measurements in the population [N]

-take the square root of the resulting value

**the standard deviation is the square root of the variance**

Variance and Standard Deviation Example:

Metabolic rates of 7 men [cal./24 hrs.]

1792 1666 1362 1614 1460 1867 1439

add them all up and divide by 7 =1600

take each observation [1792, 1666, 1362 etc.]

1792-1600=192

1666-1600=66

1362-1600=-238

1614-1600=14

1460-1600=-140

1867-1600=267

1439-1600=-161

square each of these and add them = 214, 870

variance = 214, 870 / 7 =30, 695.71

standard deviation= √30, 695.71=175.20 calories

Comparing Measures of Variability

-extreme scores affect the range the most, however standard deviation can be also affected

-sample size affects the range the most [any other measures are better choices]

-stability under sampling affects the range the most, standard deviation and variance tend to be stable

Other Comments

-when the mean is reported generally so is the standard deviation

-both variance and standard deviation are based on deviations from the mean

-when median is reported, generally so is interquartile range

-both are based on percentiles

-the range is rarely used w/any other statistic

-has no relationship to other statistics

Density Curves

-always above or on the horizontal axis

-have an area of exactly 1 underneath the curve

-area under the curve and above any range of values is the proportion of all observations that fall in that range

-the median of a density curve is the equal-areas point, the point that divides the area under the curve in half

-the mean of a density curve is the balance point, at which the curve would balance if made of solid material