1. Figure 4.6 In frequency distribution graphs, we identify the position of the mean by drawing a vertical line and labeling it with m or M. Because the standard deviation measures distance from the mean, it is represented by a line or an arrow drawn from the mean outward for a distance equal to the standard deviation and labeled with a s or an s.

2. Sample variance as an unbiased statistic
A sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter.
A sample statistic is unbiased if the average value of the statistic is equal to the population parameter.
Sample mean is an unbiased estimator of population mean
Sample variance s2 = SS/n-1
an unbiased estimator for the population variance σ2
because sample size is adjust by degrees of freedom
The average value of the statistic is obtained from all the possible samples for a specific sample size, n. see table 4.1
Table 4.1 is a math proof to show that s2 = SS/n-1 is unbiased
You will not need to do this procedure on an exam

3. Table 4.1 The set of nine samples with n = 2 selected from the population described in Example 4.7. The mean is computed for each sample, and the variance is computed two different ways: (1) dividing by n, which is incorrect and produces a biased statistic; and (2) dividing by n – 1, which is correct and produces an unbiased statistic. From a population of scores 0,0,3,3,9,9 with m = 4 and s2 = 14.
See Excel Example
36/9 = /9 = /9 = 14

4. Sample variance as an unbiased statistic
However sample standard deviation
s = √ s2 is a biased estimator of population standard deviation
but is still a good estimate of the population
Sample mean and sample variance are preferred for inferential tests
Sample mean and sample standard deviation are preferred for descriptive statistics

5. Factors that Affect Variability
Adding a constant to each score does not change the standard deviation. Mean increases by the constant so deviation scores stay the same.
S = 0.75
S = 0.75

6. Properties of the Standard Deviation
If a constant is added to every score in a distribution, the standard deviation will not be changed.
If you visualize the scores in a frequency distribution histogram, then adding a constant will move each score so that the entire distribution is shifted to a new location.
The center of the distribution (the mean) changes, but the standard deviation remains the same.
Add 5 points to each score
M = 2.5
s = 1.05
M = 7.5
s = 1.05

7. Factors that Affect Variability
Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.
S = 1.58
S = 4.02

8. Properties of the Standard Deviation
If each score is multiplied by a constant, the standard deviation will be multiplied by the same constant.
Multiplying by a constant will multiply the distance between scores, and because the standard deviation is a measure of distance, it will also be multiplied.
Multiply each score by 5
M = 2.5
s = 1.05
M = 12.5
s = 5.24

9. Transformation of scale
Because adding a constant changes the mean but not the standard deviation
Can change the scale of a set of score by adding a constant
Exam with a mean of 87
Add 13 to every score
Exam mean changes to 100, a convenient value
Because multiplying by a constant changes the mean and standard deviation
Can change the scale by multiplying by a constant
Exam with standard deviation of 5
Multiply each score by 2
Exam standard deviation changes to 10, a convenient value

10. Other Factors that Affect Variability
Extreme scores produce skewed distributions
if the population distribution is not symmetrical
should not be using mean and variance
Sample size
Samples of 10 is a better estimate then a sample size of 5
Not much benefit with sample size larger then 30

11. Mean and Standard Deviation as Descriptive Statistics
Describing an entire distribution
Rather than listing all of the individual scores in a distribution, research reports typically summarize the data by reporting only the mean and the standard deviation.
Describing the location of individual scores
The mean and the standard deviation can be used to reconstruct the underlying scale of measurement (the X values along the horizontal line).
The scale of measurement helps complete the picture of the entire distribution and helps to relate each individual score to the rest of the group.

12. Mean and Standard Deviation as Descriptive Statistics
If you are given numerical values for the mean and the standard deviation, you should be able to construct a visual image (or a sketch) of the distribution of scores.
As a general rule, about 70% of the scores will be within one standard deviation of the mean, and about 95% of the scores will be within a distance of two standard deviations of the mean.
It is common to talk about descriptive statistics as the mean plus or minus the standard deviation.
For example or as shown in figure 4.7.

13. Figure 4.7 Caution the data distribution needs to be symmetrical for this to work. Imagine what this looks like with extremely skewed data.

14. Normal Curve with Standard Deviation
| + or - one s.d. |

15. Variance and Inferential Statistics
In very general terms, the goal of inferential statistics is to detect meaningful and significant patterns in research results.
Variability plays an important role in the inferential process because the variability in the data influences how easy it is to see patterns.
In general, low variability means that existing patterns can be seen clearly, whereas high variability tends to obscure any patterns that might exist.

16. Figure Using variance in inferential statistics Graphs showing the results from two experiments. In Experiment A, the variability within samples is small and it is easy to see the 5-point mean difference between the two samples. In Experiment B, however, the 5-point mean difference between samples is obscured by the large variability within samples.

17. Reporting the Standard Deviation The number of aggressive responses of male and female children after viewing cartoons. Example of APA tale format when reporting descriptive data. Did type of cartoon make a difference on aggressive responses? How do males and females differ?

18. Chapter 5: z-Scores – Location of Scores and Standardized Distributions

19. Preview

20. 2014 Boston Marathon, Wheelchair Race
“Ernst Van Dyk (RSA) and Tatyana McFadden (USA) won the men’s and women’s titles at the 118th B.A.A. Boston Marathon. Capturing an unprecedented tenth Boston Marathon title, Van Dyk led from wire-to-wire. McFadden celebrated her 25th birthday by defending her title in the women’s race, just eight days after winning and breaking her own course record at the London Marathon.”
Using SPSS to get standardized scores

21. z-Scores and Location
One purpose of z-scores, or standard scores, is to identify and describe the exact location of each score in a distribution.
Suppose you received a score of X = 76 on a statistics exam. How did you do?
Your score of X = 76 could be one of the best scores, or it might be the lowest score.
To find the location of your score, you must have information about the other scores in the distribution.
mean for the exam
standard deviation for the exam

22. Figure 5. 2 Two distributions of exam scores
Figure 5.2 Two distributions of exam scores. For both distributions, µ = 70, but for one distribution, σ = 12. The position of X = 76 is very different for these two distributions.

23. Figure 5. 2 shown with a box for each individual score
Figure 5.2 shown with a box for each individual score. Two distributions of exam scores. For both distributions, µ = 70, but for one distribution, σ = 12. The position of X = 76 is very different for these two distributions. A B
Figure 5.2
Two distributions of exam scores. For both distributions, μ = 70, but for one distribution, σ = 3, and for the other, σ = 12. The relative position of X = 76 is very different for the two distributions.

24. z-Scores and Location
The process of changing an X value into a z-score involves creating a signed number, called a z-score, such that:
value of the z-score tells exactly where the score is located
sign of the z-score (+ or –) identifies whether the X value is located
above the mean (positive)
below the mean (negative)
numerical value of the z-score corresponds to the number of standard deviations between X and the mean of the distribution.

25. Z-scores and Locations
Process of changing an X value into a z-score
Visualize z-scores as locations in a distribution.
z = 0 is in the center (at the mean)
extreme tails
–2.00 on the left
+2.00 on the right
most of the distribution is contained between z = –2.00 and z = +2.00
z-scores identify exact locations within a distribution
z-scores can be used as
descriptive statistics
inferential statistics

26. Figure 5-3 The relationship between z-score values and locations in a population distribution.
One S.D.
Two S.D.

27. Transforming back and forth between X and z-score
The z-score formula for computing the z-score for any value of X.
z = (X – μ) ∕ σ
Example 5.3: μ = 86, σ = 7
convert X = 95 to a z-score
z = (95 – 86) ∕ 7 = 1.29

28. Transforming back and forth between X and z-score
Determining a raw score (X) from a z-score
Equation for computing the value of X corresponding to any specific z-score.
using devinitional formula
X = μ + zσ
Zσ is the deviation of X from the mean
Example:5.3: μ = 60, σ = 8
convert z = -1.5 to X value
X = 60 + (-1.5)(8)
X = = 48