Chapter 3-2 Measure of Variation

Variance

standard deviation squared

3 Important measures of Variation

1.) Range 2.) Standard Deviation 3.) Variance

Standard Deviation of a Population

A different formula is used to calculate the standard deviation ó of a population: instead of dividing by n-1 for a sample, we divide by the population size N

Standard Deviation (of a sample)

Denoted by s, is a measure of how much values deviate away from the mean s = Standard deviation ó = Population Standard deviation

Variance of a sample and a population

Is a Measure of variation equal to the square of the standard deviation Sample Variation : s^2 = Square of the standard deviation S Population Variance : ō^2 = Square of the population standard deviation ó

Range Rule of Thumb (Std. D. sample)

Is a crude but simple tool for understanding and interpreting standard deviation The vast majority (such as 95%) Of sample values lie within two standard deviations of the mean

Range

Is the difference between maximum data value and the minimum data value

Example: Find the Range of these Verizon data speeds (Mbps): 38.5, 55.6, 22.4, 14.1, & 23.1

Order Array : 14.1, 22.4, 23.1, 38.5, & 55.6 Range = Max - Min Range = 55.6 -14.1 = 41.50 (Mbps)

Empirical Rule for Data with a Bell-Shaped Distribution (normal)

The empirical rule States that for data sets having a distribution that is approximately bell shaped the following properties apply 1.) about 60% of all values fall within one standard deviation of the mean 2.) about 95% of all values fall within two standard deviation of the mean 3.) about 99.7% of all values for within three standard deviation of the mean

Important Property of Range

The range uses only the maximum and minimum data value so it is very sensitive to extreme value range is not resistant. Because the range uses only the maximum and minimum values it does not take every value into account and therefore does not truly reflect the variation among all of the data values

Important Propterties of Standard Deviation (of a sample)

The standard deviation is a measure of how much data values deviate away from the mean The value of the standard deviation ass is never negative it is zero only but all of the data values are exactly the same Larger values of "s" indicate greater amounts of variation The standard deviation S can increase dramatically with one or more outlet outliers The units of the standard deviation S (such as mins, feet, pounds) are the same as the units of the original data values The same standard deviation as is a biased estimator of the population standard deviation (ó), which means that values of the sample standard deviation ass do not center around the value (ó - population standard deviation)

Why Divide by n-1

There are only n- 1 values that can assigned without constraint. With a given mean, we can use any number for the first n-1 values, but the last value will then be automatically determined With division by n-1, sample variances s^2 tend to center around the value of the population vairance ó^2 tend to underestimate the value of the population variance ó^2

Range rule of thumb for estimating the value of the standard deviation ( s ) (Sample)

To roughly estimate the standard deviation from a collection of known sample data use S ≈ Range / 4

Round-Off Rule for Measures of Variation

When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data.