Standard Scores & Correlation

1. Standard Scores & Correlation

2. Review A frequency curve either normal or otherwise is simply a line graph of all frequency of scores earned in a data set In a normally distributed frequency curve the mean, median, and mode are all at the highest point of the curve (this happens in a normal curve, a leptokurtic curve, and a platykurtic curve) In a skewed distributed frequency curve the mode is the highest point of the curve and the mean shifts toward low extreme scores (negative) or high extreme scores (positive)

3. Review continued Variance: sum of the deviation of scores from the mean squared divided by the number of scores: it ignores direction of deviation from mean since deviation from mean is squared Standard deviation: most commonly used statistic to describe the variability of a data set The smaller the standard deviation the more scores are clustered near the mean The larger the standard deviation the more spread out scores are from the mean

4. Standard Scores • Allow meaningful comparisons to be made between different sets of data. • In other words they provide us with a commonality with which to compare multiple types of scores • Percentile Rank • z score • T score

5. Percentile Rank Ordinal level of data 50th percentile is by definition the median Calculated from a simple frequency or group frequency distribution table

6. Calculating Percentile Rank Formula for Percentile Rank: PR for X =[∑fb + ((X - |r|) (fw))] (100) i N

7. What does that all mean? PR= percentile rank (PRx is the percentile rank of a particular score) ∑fb= cumulative frequency of the scores in the interval below the interval containing the score X= score to find the percentile rank for /r/= real lower limit of the interval containing the score fw= frequency of the interval containing the score int= the interval size N= total number of scores in the data set

8. Let’s try one! Group X f cf c% 23-25 3 66 100 20-22 7 63 95 17-19 15 56 85 14-16 19 41 62 11-13 11 22 33 8-10 9 11 14 5-7 2 2 1

9. Let’s try another one: Group X f cf c% 90-86 3 35 100 85-81 10 32 91 80-76 10 22 63 75-71 7 12 34 70-66 3 5 14 65-61 2 2 6

10. Calculating percentiles Formula for calculating percentiles %ile = /r/ + .X(N) - ∑fb(int) fw

11. What does that all mean? %ile=raw score that corresponds to a given percentile rank. ∑fb= cumulative frequency of the scores in the interval below the interval containing the score .X= the percentile to find (50th or 80th percentiles for example) /r/= real lower limit of the interval containing the score we are looking for fw= frequency of the interval containing the score int= the interval size N= total number of scores in the data set

12. Let’s give it a try Interval f cf 80-82 1 60 77-79 3 59 74-76 5 56 71-73 6 51 68-70 9 45 65-67 12 36 62-64 9 24 59-61 6 15 56-58 5 9 53-55 3 4 50-52 1 1 X.50? X.80?

13. z scores Standard score expressed in terms of standard deviation units which indicates distance raw score is from mean. z scores can be positive (score above mean) or negative (score below mean)* A z score of 0 is the mean z= X-X or X-X * s s * occurs when the better score is lower than the mean (golf, time in a race, percentage of body fat)

14. Let’s try it: You have a set of scores for the long jump and the mean is 50 with a standard deviation of 5. What would the z score of the score 42 be? How about 55? How about this one: you have a set of golf scores with a mean of 10 and a standard deviation of 2. What would the z score of the score 12 be? How about 7?

15. T scores Derived from a z score Will always be a positive whole number with a T score of 50 representing the mean Easier for some to comprehend since it is on a scale of 0-100 T= 10z +50 or 10 (x-x) +50 s So, can you calculate a T score from a -2.4 z score? How about from a z score of +3.7?

16. z and T scores z scores have a standard deviation of 1 with a mean of 0 T scores have a standard deviation of 10 with a mean of 50 So, a +1 z score = a T score of 60; a -2 z score = a T score of 30 Since we know that when scores are normally distributed that 99.7% of all scores will fall between a standard deviation of +- 3, we rarely have a T score above 80 or below 20 We can also so that a score at the 84th percentile is equal to a z score of +1 and a T score of 60. Why?

17. Correlation Relationship between two variables Correlation coefficient is the statistic that indicates the relationship or association between variables Correlation does not mean cause and effect You could compare the relationship between height and weight for a group of students (+ correlation) You could compare speed in the 100 meters and long jump (inverse correlation)

18. Correlation Coefficients With a positive correlation the change of direction of both variables will be the same (either both increase or decrease) With an inverse correlation the change of direction of one variable increases as the other decreases The degree of correlation or relationship is determined by a number from -1.00 to +1.00 The higher the number regardless of sign, the more closely related the variables The lower the number regardless of sigh, the less closely related the variables

19. Rules to remember • Correlation coefficients fall between +- 1.00 • The sign of a correlation coefficient indicates type of relationship (positive or inverse) • +- .80- 1.00= high relationship • +- .60-.79 = Moderately high relationship • +- .40-.59 = Moderate relationship • +- .20- .39 = Low relationship • +-0-.19 = no relationship • +.75 and -.75 indicate the same degree of relationship but one is positive and one is inverse

20. Correlational Procedures • Pearson Product Moment Correlation …..requires interval or ratio data • Spearman Rho Rank Order Correlation …..requires ordinal data