Friday, October 2 Variability

CENTRAL TENDENCY Mode Median Mean nominal ordinal interval

Variability CENTRAL TENDENCY Mode Median Mean nominal ordinal interval Range Interquartile Range “Then there is the man who drowned crossing a stream with an average depth of six inches.” - W.I.E. Gates interval Variance Standard Deviation

Range

Box Plot Interquartile Range Range

_ X

The population mean is µ. The sample mean is X. _ _ The population mean is µ. The sample mean is X.

s Population µ  _ X _ The population mean is µ. The sample mean is X. The population standard deviation is , the sample sd is s.

SS 2 N Variance of a population, 2 (sigma squared). It is the sum of squares divided (SS) by N SS 2 = N

SS  (X –  ) 2 N Variance of a population, 2 (sigma squared). It is the sum of squares divided (SS) by N  (X –  ) 2 SS 2 = N

SS  N The Standard Deviation of a population,  It is the square root of the variance. SS  = N This enables the variability to be expressed in the same unit of measurement as the individual scores and the mean.

The population mean is µ. The sample mean is X. _ _ The population mean is µ. The sample mean is X.

s Population µ  _ X _ The population mean is µ. The sample mean is X. The population standard deviation is , the sample sd is s.

In reality, the sample mean is just one of many possible sample SampleC XC _ SampleD XD _ Population SampleB XB _ µ SampleE XE SampleA XA _ _ In reality, the sample mean is just one of many possible sample means drawn from the population, and is rarely equal to µ.

In reality, the sample mean is just one of many possible sample SampleC XC _ SampleD XD sc _ sd Population SampleB XB _ µ  sb SampleE XE SampleA XA _ _ se sa In reality, the sample mean is just one of many possible sample means drawn from the population, and is rarely equal to µ.

Sampling error = Statistic - Parameter _ Sampling error for the mean = X - µ Sampling error for the standard deviation = s - 

Unbiased and Biased Estimates An unbiased estimate is one for which the mean sampling error is 0. An unbiased statistic tends to be neither larger nor smaller, on the average, than the parameter it estimates. The mean X is an unbiased estimate of µ. The estimates for the variance s2 and standard deviation s have denominators of N-1 (rather than N) in order to be unbiased. _

SS 2 = N

SS s2 = (N - 1)

_  (X – X ) 2 SS s2 = (N - 1)

SS (N - 1) s =

Computational formula Conceptual formula VS Computational formula

What is a measure of variability good for?