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