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Chapter 3

Chapter 3. The Normal Curve. Where have we been?.  = = 1.79. (X- ) = 0.00. (X- ) 2 = SS = 16.00. X = 30 N = 5  = 6.00.

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Chapter 3

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  1. Chapter 3 The Normal Curve

  2. Where have we been?

  3.  = = 1.79 (X- ) = 0.00 (X- )2 = SS = 16.00 X = 30 N = 5  = 6.00 To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum them (SS), divide by N (2) and take a square root(). Example: Scores on a Psychology quiz Student John Jennifer Arthur Patrick Marie X 7 8 3 5 7 X -  +1.00 +2.00 -3.00 -1.00 +1.00 (X - )2 1.00 4.00 9.00 1.00 1.00 2 = SS/N = 3.20

  4. Stem and Leaf Display • Reading time data Reading Time 2.9 2.9 2.8 2.8 2.7 2.7 2.6 2.6 2.5 2.5 Leaves 5,5,6,6,6,6,8,8,9 0,0,1,2,3,3,3 5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,9 0,0,1,2,3,3,3,3,4,4,4 5,5,5,5,6,6,6,8,9,9 0,0,0,1,2,3,3,3,4,4 5,6,6,6 0,1,1,1,2,3,3,4 6,6,8,8,8,8,8,9,9,9 0,1,1,1,2,2,2,4,4,4,4 i = .05 #i = 10

  5. Transition to Histograms 9 9 9 9 7 7 7 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 2 2 2 1 1 1 0 4 4 4 3 3 3 3 2 1 0 0 9 9 9 8 8 8 8 8 6 6 4 4 3 3 3 2 1 0 0 0 9 9 8 6 6 6 5 5 5 5 9 8 8 6 6 6 6 5 5 4 3 3 2 1 1 1 0 3 3 3 2 1 0 0 6 6 6 5

  6. Histogram of reading times 20 18 16 14 12 10 8 6 4 2 0 F r e q u e n c y Reading Time (seconds)

  7. Normal Curve

  8. Principles of Theoretical Curves • Expected frequency = Theoretical relative frequency * N • Expected frequencies are your best estimates because they are closer, on the average, than any other estimate when we square the error. • Law of Large Numbers - The more observations that we have, the closer the relative frequencies should come to the theoretical distribution.

  9. The Normal Curve • Described mathematically by Gauss in 1851. So it is also called the “Gaussian”distribution. It looks something like a bell, so it is also called a “bell shaped” curve. • The normal curve really represents a histogram whose rectangles have their corners shaved off with calculus. • The normal curve is symmetrical. • The mean (mu) falls exactly in the middle. • 68.26% of scores fall within 1 standard deviation of the mean. • 95.44% of scores fall within 2 standard deviations of the mean. • 99.74% of scores fall within 3 standard deviations of mu.

  10. The normal curve and Z scores • The normal curve is a theoretical distribution that underlies most variables that are of interest to psychologists. • A Z score expresses the number of standard deviations that a score is above or below the mean in a normal distribution. • Any point on a normal curve can be referred to with a Z score

  11. The Z table and the curve • The Z table shows the normal curve in tabular form as a cumulative relative frequency distribution. • That is, the Z table lists the proportion of a normal curve between the mean and points further and further from the mean. • The Z table shows only the cumulative proportion in one half of the curve. The highest proportion possible on the Z table is therefore .5000

  12. Why does the Z table show cumulative relative frequencies only for half the curve? • The cumulative relative frequencies for half the curve are all one needs for all relevant calculations. • Remember, the curve is symmetrical. • So the proportion of the curve between the mean and a specific Z score is the same whether the Z score is above the mean (and therefore positive) or below the mean (and therefore negative). • Separately showing both sides of the curve in the Z table would therefore be redundant and (unnecessarily) make the table twice as long.

  13. KEY CONCEPT The proportion of the curve between any two points on the curve represents the relative frequency of scores between those points.

  14. With a little arithmetic, using the Z table, we can determine: The proportion of the curve above or below any Z score. Which equals the proportion of the scores we can expect to find above or below any Z score. The proportion of the curve between any two Z scores. Which equals the proportion of the scores we can expect to find between any two Z scores.

  15. The mean One standard deviation Standard deviations 3 2 1 0 1 2 3 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z scores Normal Curve – Basic Geography F r e q u e n c y Measure |---34.13--|--34.13---| Percentages |--------47.72----------|----------47.72--------| |--------------49.87-----------------|------------------49.87------------|

  16. The z table Z Score 0.00 0.01 0.02 0.03 0.04 . 1.960 2.576 . 3.90 4.00 4.50 5.00 Proportion mu to Z .0000 .0040 .0080 .0120 .0160 . .4750 .4950 . .49995 .49997 .499997 .4999997 The Z table contains a column of Z scores coordinated with a column of proportions. The proportion represents the area under the curve between the mean and any other point on the curve. The table represents half the curve See pages 54 and 55

  17. Common Z scores – memorize these scores and proportions Z Proportion Score mu to Z 0.00 .0000 1.00 .3413 2.00 .4772 3.00 .4987 1.960 .4750 ( * 2 = 95% between Z= –1.960 and Z= +1.960) (* 2 = 99% between Z= –2.576 and Z= + 2.576) 2.576 .4950

  18. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE - Proportion between a score and the mean. F r e q u e n c y Proportion mu to Z for -0.30 = .1179 Proportion score to mean =.1179 score 470 .

  19. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE - Proportion between score F r e q u e n c y Proportion mu to Z for -0.30 = .1179 Proportion between +Z and -Z = .1179 + .1179 = .2358 score 470 530 .

  20. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE – Proportion of the curve above a score. F r e q u e n c y Proportion mu to Z for .30 = .1179 Proportion above score = .1179 + .5000 = .6179 score 470 Proportion above score.

  21. Proportion mu to Z for -1.06 = .3554 Proportion mu to Z for .37 = .1443 +0.37 -1.06 Area Area Add/Sub Total Per Z1Z2mu to Z1mu to Z2Z1 to Z2AreaCent -1.06 +0.37 .3554 .1443 Add .4997 49.97 % USING THE Z TABLE - Proportion between score and a different point on the other side of the mean. F r e q u e n c y -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z scores Percent between two scores.

  22. Proportion mu to Z for 1.12 = .3686 +1.12 +1.50 Area Area Add/Sub Total Per Z1Z2mu to Z1mu to Z2Z1 to Z2AreaCent +1.50 +1.12 .4332 .3686 Sub .0646 6.46 % USING THE Z TABLE - Proportion between score and another point on the same side of the mean. Proportion mu to Z for 1.50 = .4332 F r e q u e n c y -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z scores Percent between two scores.

  23. F r e q u e n c y Proportion mu to Z for -0.30 = .1179 Standard deviations 470 3 2 1 0 1 2 3 USING THE Z TABLE – Expected frequency = theoretical relative frequency * number of participants (EF=TRF*N).Expected frequency between mean and Z = -.30. If N = 300. EF= .1179*300 = 35.37 .

  24. F r e q u e n c y Proportion mu to Z for .30 = .1179 Proportion above score = .1179 + .5000 = .6179 Standard deviations -.30 3 2 1 0 1 2 3 USING THE Z TABLE – Expected frequency = theoretical relative frequency * number of participants (EF=TRF*N).Expected frequency above Z = -.30 if N = 300.EF=.6179 * 300 = 185.37

  25. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE – Percentage below a score F r e q u e n c y Percentage = 50 % up to mean + 34.13% for 1 SD = 84.13% inches What percent of the population scores at or under a Z score of +1.00

  26. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE – Percentile Rank is the proportion of the population you score as well as or better than times 100. F r e q u e n c y Percentile: .5000 up to mean + .3413 =.8413 .8413 * 100 =84.13 =84th percentile inches What is the percentile rank of someone with a Z score of +1.00

  27. Percentile rank is the proportion of the population you score as well as or better than times 100. The proportion you score as well as or better than is shown by the part of the curve to the left of your score.

  28. Computing percentile rank • Above the mean, add the proportion of the curve from mu to Z to .5000. • Below the mean, subtract the proportion of the curve from mu to Z from .5000. • In either case, then multiply by 100 and round to the nearest integer (if 1st to 99th). • For example, a Z score of –2.10 • Proportion mu tg Z = .4821 • Proportion at or below Z = .5000 - .4821 =.0179 • Percentile = .0179 * 100 = 1.79 = 2nd percentile

  29. A rule about percentile rank • Between the 1st and 99th percentiles, you round off to the nearest integer. • Below the first percentile and above the 99th, use as many decimal places as necessary to express percentile rank. • For example, someone who scores at Z=+1.00 is at the 100(.5000+.3413) = 84.13 = 84th percentile. • Alternatively, someone who scores at Z=+3.00 is at the 100(.5000+.4987)=99.87= 99.87th percentile. Above 99, don’t round to integers. • We never say that someone is at the 0th or 100th percentile.

  30. Calculate percentiles Z Area Add to .5000 (if Z > 0) Proportion Percentile Scoremu to ZSub from .5000 (if Z < 0)at or below -2.22 .4868 .5000 - .4868 .0132 1st -0.68 .2517 .5000 - .2517 .2483 25th +2.10 .4821 .5000 + .4821 .9821 98th +0.33 .1293 .5000 + .1293 .6293 63rd +0.00 .0000 .5000 +- .0000 .5000 50th

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