Sum of Squares Example 3
Remember that the sum of squares equation is .
Consider weights, in pounds, for 6 men: 180, 192, 165, 164, 197, 192
SS can be computed as follows:
(180 -181.67)² = 2.79
(192 – 181.67)² =106.71
(165 – 181.67)²=277.89
(164 – 181.67)²=312.23
(197 – 181.67)²=235.01
(192 – 181.67)² =106.71
SS = 1041.34 squared deviations about
Now find the Variance. Remember that so or = 208.27squared lbs.
Now find the Standard Deviation (s). Remember that so = 14.43 lbs. Please notice how the Standard Deviation has now been reduced back into lbs. or the original units of “measure.” This is why it is the most user friendly statistic for dispersion. The scores are spread out, on average about the mean, by approximately 14 pounds. Doesn’t seem like a lot of variability in the scores.
Random Sampling: Every person or observation has an equal chance/probability of being selected.
Sampling Problem / Example
Given that (Population Mean)µ = 180 lbs. and (Total Population)N = 100,000
Draw several samples and compute the mean:
n=50 = 178n=60 = 170 (Please note that varies from sample to sample. This is
n=50 = 192n=50 = 179 called “sampling variability” or “sampling error”)
n=75 = 181
Simply because the sample mean or varies from sample to sample does not mean the population mean (µ) changes. The more samples you take or the larger the sample sizes you use the closer you get to the actual µ (Population Mean). “In the long run” the average of the different values would approach the µ. This is what statisticians mean when they refer to the sample mean () as an “unbiased estimate” (or “unbiased estimator”) of the population mean (µ).
Different symbols for sample statistics and population parameters
Sample Population
µ
n N
Unbiased vs. Biased
As stated earlier, is an unbiased estimate of µ. In the same regard, is an unbiased estimate of the population standard deviation, σ (lowercase sigma). The biased estimate of σ would be . Please note that you go from “s” to “S” and in the denominator it goes from “n-1” to simply “n”. What happens when you use the biased estimator is that you will often undershoot the actual population standard deviation. In other words, the value for S will often be too low. Please recall our standard deviation in Example 3. In Example 3 s= 14.43. If you used the biased estimator vs. the unbiased estimator your deviation would differ.
For Example: or = = 13.17 lbs. Notice how the value is lower than the unbiased estimate. It is also likely to be lower than the population standard deviation (which we don’t know for the example) since it is a biased estimate.
****Quizam #1 on 01/25/2011**** Practice computing mean, median, mode; practice the last lab assignment; study your notes to prepare for multiple-choice items. The quizam will be one sheet, front-and-back.