3.2 Measures of Dispersion

D ATA ● Comparing two sets of data ● The measures of central tendency (mean, median, mode) measure the differences between the “average” or “typical” values between two sets of data ● The measures of dispersion in this section measure the differences between how far “spread out” the data values are

R ANGE ● The range of a variable is the largest data value minus the smallest data value ● Example…Compute the range of 6, 1, 2, 6, 11, 7, 3, 3

R ANGE ● The range only uses two values in the data set – the largest value and the smallest value ● The range is not resistant ● If we made a mistake and 6, 1, 2 was recorded as 6000, 1, 2 Find the Range of both sets of Data

V ARIANCE ● The variance is based on the deviation from the mean  ( x i – μ ) for populations  ( x i – ) for samples ● To treat positive differences and negative differences, we square the deviations  ( x i – μ ) 2 for populations  ( x i – ) 2 for samples

P OPULATION V ARIANCE ● The population variance of a variable is the sum of these squared deviations divided by the number in the population ● The population variance is represented by σ 2 ● Note: For accuracy, use as many decimal places as allowed by your calculator

P OPULATION V ARIANCE ● Compute the population variance of 6, 1, 2, 11 ● Compute the population mean first ● Now compute the squared deviations ● Average the squared deviations ● The population variance σ 2 is ?

S AMPLE V ARIANCE ● The sample variance of a variable is the sum of these squared deviations divided by one less than the number in the sample ● The sample variance is represented by s 2 ● We say that this statistic has n – 1 degrees of freedom

S AMPLE V ARIANCE ● Compute the sample variance of 6, 1, 2, 11 ● Compute the sample mean first ● Now compute the squared deviations ● Average the squared deviations (Remember…n-1) ● The sample variance s 2 is ?

D IFFERENCE Why are the population variance and the sample variance different for the same set of numbers? Why do we use different formulas? The reason is that using the sample mean is not quite as accurate as using the population mean If we used “ n ” in the denominator for the sample variance calculation, we would get a “biased” result

S TANDARD D EVIATION ● The standard deviation is the square root of the variance ● The population standard deviation  Is the square root of the population variance ( σ 2 )  Is represented by σ ● The sample standard deviation  Is the square root of the sample variance ( s 2 )  Is represented by s

S TANDARD D EVIATION ● If the population is { 6, 1, 2, 11 }  The population variance σ 2 = 15.5  The population standard deviation σ = ● If the sample is { 6, 1, 2, 11 }  The sample variance s 2 = 20.7  The sample standard deviation s = ● The population standard deviation and the sample standard deviation apply in different situations

W HY S TANDARD D EVIATION ? The standard deviation is very useful for estimating probabilities

E MPIRICAL R ULE ● The empirical rule ● If the distribution is roughly bell shaped, then  Approximately 68% of the data will lie within 1 standard deviation of the mean  Approximately 95% of the data will lie within 2 standard deviations of the mean  Approximately 99.7% of the data (i.e. almost all) will lie within 3 standard deviations of the mean

E MPIRICAL R ULE ● For a variable with mean 17 and standard deviation 3.4  Approximately 68% of the values will lie between (17 – 3.4) and ( ), i.e and 20.4  Approximately 95% of the values will lie between (17 – 2  3.4) and (  3.4), i.e and 23.8  Approximately 99.7% of the values will lie between (17 – 3  3.4) and (  3.4), i.e. 6.8 and 27.2 ● A value of 2.1 (less than 6.8) and a value of 33.2 (greater than 27.2) would both be very unusual

D RAWING THE P ICTURE I S H ELPFUL !

C ALCULATOR How Can My Calculator Help Me??? 1 Var-Stats Remember… Greek Letters are Population

E XAMPLE 3.2 Handout