Standard Scores. Richard S. Balkin, Ph.D., LPC-S, NCC
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1 Standard Scores Richard S. Balkin, Ph.D., LPC-S, NCC 1
2 Normal Distributions While Best and Kahn (2003) indicated that the normal curve does not actually exist, measures of populations tend to demonstrate this distribution It is based on probability the chance of certain events occurring R. S. Balkin,
3 Normal Curve The curve is symmetrical 50% of scores are above the mean, 50% below The mean, median, and mode have the same value Scores cluster around the center R. S. Balkin,
4 Normal distribution " " rule One standard deviation away from the mean in either direction (red) includes about 68% of the data values. Another standard deviation out on both sides includes about 27% more of the data (green). The third standard deviation out adds another 4% of the data (blue). R. S. Balkin,
5 The Normal Curve R. S. Balkin,
6 Interpretations of the normal curve Percentage of total space included between the mean and a given standard deviation (z distance from the mean) Percentage of cases or n that fall between a given mean and standard deviation Probability that an event will occur between the mean and a given standard deviation Calculate the percentile rank of scores in a normal distribution Normalize a frequency distribution Test the significance of observed measures in an experiment R. S. Balkin,
7 Interpretations of the normal curve The normal curve has 2 important pieces of information We can view information related to where scores fall using the number line at the bottom of the curve. I refer to this as the score world It can be expressed in raw scores or standard scores scores expressed in standard deviation units We can view information related to probability, percentages, and placement under the normal curve. I refer to this as the area world R. S. Balkin,
8 Interpretations of the normal curve Area world Score world R. S. Balkin,
9 Does the normal curve really exist? the normal distribution does not actually exist. It is not a fact of nature. Rather, it is a mathematical model an idealization that can be used to represent data collected in behavioral research (Shavelson, 1996, p. 120). R. S. Balkin,
10 Does the normal curve really exist? Glass & Hopkins (1996): God loves the normal curve (p. 80). No set of empirical observations is ever perfectly described by the normal distribution but an independent measure taken repeatedly will eventually resemble a normal distribution Many variables are definitely not normally distributed (i.e. SES) The normal curve has a smooth, altogether handsome countenance a thing of beauty (p.83). R. S. Balkin,
11 Nonnormal distributions Positively skewed majority of the scores are near the lower numbers Negatively skewed the majority of the scores are near the higher numbers Bimodal distributions have two modes R. S. Balkin,
12 Positively skewed If a test was very difficult and almost everyone in the class did very poorly on it, the resulting distribution would most likely be positively skewed. In the case of a positively skewed distribution, the mode is smaller than the median, which is smaller than the mean. The mode is the point on the x-axis corresponding to the highest point, that is the score with greatest value, or frequency. The median is the point on the x-axis that cuts the distribution in half, such that R. S. Balkin, % of the area falls on each side. The mean is pulled by the extreme scores on the right.
13 Negatively skewed A negatively skewed distribution is asymmetrical and points in the negative direction, such as would result with a very easy test. On an easy test, almost all students would perform well and only a few would do poorly. The order of the measures of central tendency would be the opposite of the positively skewed distribution, with the mean being smaller than the median, which is smaller than the mode. R. S. Balkin,
14 Normal Curve Summary Unimodal Symmetry Points of inflection Tails that approach but never quite touch the horizontal axis as they deviate from the mean R. S. Balkin,
15 Standard scores Standard scores assume a normal distribution They provide a method of expressing any score in a distribution in terms of its distance from the mean in standard deviation units Z score T score R. S. Balkin,
16 Z-score A raw score by itself is rather meaningless. What gives the score meaning is its deviation from the mean A Z score expresses this value in standard deviation units R. S. Balkin,
17 Z-score X " X x z = or!! X = Raw score X! x = = = Mean Standard deviation ( X! X ) R. S. Balkin,
18 Z score formula z = X "! X or x! X = 76 X = 82! = 4 R. S. Balkin,
19 Z score computation z = X "! X or x! X X = = 76 82! = 4 76! 82! 6 z = = =! R. S. Balkin,
20 T score Another version of a standard score Converts from Z score Avoids the use of negative numbers and decimals R. S. Balkin,
21 T score formula T=50+10z Always rounded to the nearest whole number A z score of 1.27= R. S. Balkin,
22 T score formula T=50+10z Always rounded to the nearest whole number A z score of 1.27= T=50+10(1.27)= R. S. Balkin,
23 T score formula T=50+10z Always rounded to the nearest whole number A z score of 1.27= T=50+10(1.27)= =62.70=63 R. S. Balkin,
24 More on standard scores Any standard score can be converted to standard deviation units A test with a mean of 500 and standard deviation of 100 would be R. S. Balkin,
25 More on standard scores Any standard score can be converted to standard deviation units A test with a mean of 500 and standard deviation of 100 would be x = z! R. S. Balkin,
26 Calculating the distribution that fall before, between, or beyond the mean and standard deviation From this point on represents above z, 16% From this point and below represents z b, 84 th percentile z a R. S. Balkin,
27 Confidence Intervals Confidence intervals provide a range of values given error in a score For example, if the mean = 20 then we can be 68% confidence that the score will be plus or minus 1 sd from the mean 95% confidence that the score will be plus or minus 2 sd from the mean 99% confidence that the score will be plus or minus 3 sd from the mean R. S. Balkin,
28 Confidence intervals A mean of 20 with a sd of 5 68% confidence that the score will be between 15 to 25 95% confidence that the score will be between 10 to 30 99% confidence that the score will be between 5 to 35 R. S. Balkin,
29 Confidence Intervals A mean of 48 and a sd of % confidence that the score will be between to 95% confidence that the score will be between to 99% confidence that the score will be between to R. S. Balkin,
30 Confidence Intervals A mean of 48 and a sd of % confidence that the score will be between to % confidence that the score will be between to 99% confidence that the score will be between to R. S. Balkin,
31 Confidence Intervals A mean of 48 and a sd of % confidence that the score will be between to % confidence that the score will be between 42.5 to % confidence that the score will be between to R. S. Balkin,
32 Confidence Intervals A mean of 48 and a sd of % confidence that the score will be between to % confidence that the score will be between 42.5 to % confidence that the score will be between to R. S. Balkin,
33 Correlation Relationship between two or more paired variables or two or more data sets Correlation = r or p Correlations range from (perfect negative correlation) to (perfect positive correlation) A perfect correlation indicates that for every unit of increase (or decrease) in one variable there is an increase (or decrease) in another variable R. S. Balkin,
34 Positive correlation R. S. Balkin,
35 Negative correlation R. S. Balkin,
36 Low correlation R. S. Balkin,
37 Types of correlations Pearson s Product-Moment Coefficient of correlation Known as a Pearson s r Most commonly used Spearman Rank order coefficient of correlation Known as Spearman rho (p) Only utilized with ordinal values R. S. Balkin,
38 Interpreting a correlation coefficient Be aware of outliers scores that differ markedly from the rest of the sample Look at direction (positive or negative) and magnitude (actual number) Does not imply cause and effect See the table on p. 388 for interpreting a correlation coefficient R. S. Balkin,
39 Interpreting a correlation coefficient Using the table on p. 388 interpret the following correlation coefficients: R. S. Balkin,
40 Interpreting a correlation coefficient Using the table on p. 388 interpret the following correlation coefficients: +.52 Moderate -.78 Substantial +.12 Negligible R. S. Balkin,
41 R. S. Balkin, Computing a Pearson r ) )( ( ) ( ) ( ) )( ( Y Y X X xy Y Y y X X x xy y x r!! = "! = "! = " " " " = # # #
42 Computing Pearson r X Y x 2 x y 2 y xy R. S. Balkin,
43 Computing Pearson r X Y x 2 x y 2 y xy mean = 6 5 sum=24 sum = R. S. Balkin,
44 Computing Pearson r r = 23 (24)(28) R. S. Balkin,
45 Computing Pearson r 23 r = (24)(28) r =.8872 R. S. Balkin,
46 Correlational Designs We use correlations to explore relationships between two variables We can also use a correlation to predict an outcome In statistics this is known as a regression analysis R. S. Balkin,
47 Correlational Designs For example, a correlation of =.60 means that for every increase in X there is a.60 standard deviation unit increase in Y Correlational designs are different from experimental designs In a correlational design, we explore relationships between two or more variables that are interval or ratio In a correlational design we do not compare groups; we do not have random assignment (though we do have random sampling) R. S. Balkin,
48 Correlational Designs For example, maybe we want to know the relationship between self-esteem and depression. We could use two instruments, one that measures self-esteem and one that measures depression. Then we can conduct a regression analysis and see if the scores on one instrument predict scores on the other I would hypothesize that high scores in depression correlation with low scores in selfesteem. R. S. Balkin,
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