More basics: central tendency, variability, populations and samples. Monday, October 3 More basics: central tendency, variability, populations and samples.

The mode is the score with the highest frequency of occurrences. It is the easiest score to spot in a distribution. It is the only way to express the central tendency of a nominal level variable.

The median. The median is the middle-ranked score (50th percentile). If there is an even number of scores, it is the arithmetic average of the two middle scores. The median is unchanged by outliers. Even if Bill Gates were deleted from the U.S. economy, the median asset of U.S. citizens would remain (more or less) the same.

 _ Xi X The Mean The mean is the arithmetic average of the scores. _________ i X = N

 _ Xi X The Mean The mean is the arithmetic average of the scores. The mean is the center of gravity of a distribution. Deleting Bill Gates’ assets would change the national mean income. _  Xi _________ i X = N

The mean of a group of scores is that point on the number line such that the sum of the squared distances of all scores to that point is smaller than the sum of the squared distances to any other point.

The Mean The sum of squared deviations from the Mean is at the lowest value. _ ( ) 2  Xi - X is lowest

The Mean The sum of squared deviations from the Mean is at the lowest value. _ ( ) 2  Xi - X is lowest _ X

The Mean The mean is the arithmetic average of the scores. The mean is the center of gravity of a distribution. Deleting Bill Gates’ assets would change the national mean! The sum of squared deviations from the Mean is at the lowest value. The mean is not a good measure of central tendency if there are outliers.

Variability

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 =