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

3. 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.

4. 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.

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

6.  _ 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

7. 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.

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

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

10. 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.

13. Variability

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

15. 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

16. 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.

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

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

19. In reality, the sample mean is just one of many possible sample
SampleC XC _
SampleD XD _
Population
SampleB XB _
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.

20. In reality, the sample mean is just one of many possible sample
SampleC XC _
SampleD XD sc _ sd
Population
SampleB XB _  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.

21. Sampling error = Statistic - Parameter_
Sampling error for the mean = X - µ
Sampling error for the standard deviation = s - 

22. 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. _

23. SS 2 = N

24. SS s2 =
(N - 1)

25. _
 (X – X ) 2 SS s2 =
(N - 1)

26. SS
(N - 1) s =