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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. The variance and standard deviation are numbers that describe how far, on the average, scores are from their mean, mu. • But we often want additional detail about how scores will fall around their mean. • We may also wish to theorize about how scores should fall around their mean.

  5. Describing and theorizing about how scores fall around their mean. • Frequency distributions • Stem and leaf displays • Bar graphs and histograms • Theoretical frequency distributions

  6. Absolute Frequency 117 157 158 115 78 44 21 7 6 1 3 1 708 Frequency distributions Cumulative Relative Frequency .165 .387 .610 .773 .883 .945 .975 .983 .993 .994 .999 1.000 Cumulative Frequency 117 274 432 547 625 669 690 697 703 704 707 708 Cumulative frequencies show number of scores at or below each point. # of acdnts 0 1 2 3 4 5 6 7 8 9 10 11 Calculate by adding all scores below each point. Cumulative relative frequencies show the proportion of scores at or below each point. Calculate by dividing cumulative frequencies by N at each point.

  7. 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

  8. 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

  9. 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)

  10. Normal Curve

  11. Principles of theoretical frequency distributions • Expected frequency = Theoretical relative frequency X 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.

  12. Using the theoretical frequency distribution known as the normal curve

  13. 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 is a figural representation of a theoretical frequency distribution. • The frequency distribution represented by 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.

  14. IMPORTANT CONCEPT: Since the curve is symmetrical around the mean, whatever happens on one side of the curve is exactly mirrored on the other side.

  15. The normal curve and Z scores • The normal curve is the theoretical relative frequency 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

  16. 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

  17. Why does the Z table show cumulative relative frequencies only for half the curve?

  18. 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.

  19. IMPORTANT CONCEPT: The proportion of the curve between any two points on the curve represents the theoretical relative frequency (TRF) of scores between those points.

  20. Area of the curve between two points • If the area of the curve between two points is 56.32% of the curve, we would expect to find a proportion of .5632 of the scores between those two points.

  21. 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.

  22. 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------------|

  23. 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 pairs of columns: columns of Z scores coordinated with columns of proportions from mu to Z. The columns of proportions show the proportion of the scores that can be expected to lie between the mean and any other point on the curve. The Z table shows the cumulative relative frequencies for half the curve. See pages 54 and 55

  24. Proportion = .1293 + .1293 = .2586 Another important concept: Most scores are close to the mean! So if you have two equal sized intervals, the one closer to mu contains a higher proportion of scores What proportion of scores falls in the interval between Zs of -.50 to +.50 (an interval of one standard deviation right around the mean)? .1915 + .1915 = .3830 (almost 40%) Note: this is the one-standard-deviation-wide interval with the highest proportion anywhere on the curve.

  25. Intervals further from mu What proportion of scores falls in the interval between Zs of 0.00 to +1.00 (an interval of one standard deviation starting at the mean, but not right around it)? This one can be read directly from the table - .3413 (It is a little over a third) What proportion of scores falls in the interval between Zs of +.50 to +1.50 (an interval of one standard deviation a little further from the mean)? ..4332 - .1915 = .2417 This time we are down to less than a quarter of the population.

  26. 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 ( x 2 = 95% between Z= –1.960 and Z= +1.960) (x 2 = 99% between Z= –2.576 and Z= + 2.576) 2.576 .4950

  27. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE - Proportion of the scores between a specific Z 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 .

  28. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE - Proportion of the scores in a population between two Z scores that are identical in size, but have opposite signs. 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 .

  29. The critical values of the normal curve • Critical values of a distribution show which symmetrical interval around mu contains 95% and 99% of the curve. • In the Z table, the critical values are starred and shown to three decimal places • 95% (a proportion of .9500) is found between Z scores of –1.960 and +1.960 • 99% (a proportion of .9900) is found between Z scores of –2.576 and +2.576

  30. Standard deviations 3 2 1 0 1 2 3 USING THE Z TABLE – Proportion of the scores in a population above a specific Z 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

  31. 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 of scores between a two different Z scores on opposite sides of the mean. (ADD THE TWO PROPORTIONS!). 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.

  32. 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 of scores between two Z scores on the same side of the mean. (Subtract the smaller proportion from the larger one.) 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.

  33. What proportion of the curve falls between a Z of -.30 and +1.30? • What proportion of the curve falls between a Z of +.30 and +1.30?

  34. Add the proportions if Z scores are on opposite sides of the mean, subtract if the scores are on the same side • Between the mean and a Z of -.30 there is a proportion of .1179. • Between the mean and a Z of +1.30 there is a proportion of .4032 • Between -.30 and +1.30 there is a proportion of .1179 +.4032 = .5211 • Between +.30 and +1.30 there is a proportion of .4032-.1179 = .2853

  35. Obtaining expected frequencies (EF) from the normal curve. • Basic rule: To find an expected frequency, multiply the proportion of scores expected in the part of the curve by the total N. Expected frequency = theoretical relative frequency x N.

  36. Expected frequencies are another least squared, unbiased prediction. Expected frequencies usually must be wrong, as they are routinely written to two decimal places, while it is impossible to actually find 65 hundreths of a score anywhere. So, expected frequencies are another set of least squared, unbiased predictions. Such predictions can be expected to be wrong, but close.

  37. EF=TRF x N • In the examples that I’ve solved that follow, let’s assume we have a population of size 300 (N=300) • To find the expected frequency, compute the proportion of the curve between two specific Z scores, just as we have been doing. • Then multiply that proportion (also called the theoretical relative frequency or TRF) by N.

  38. 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 Expected frequency = theoretical relative frequency x number of participants (EF=TRF*N).TRF from mean to Z = -.30 = .1179.If N = 300: EF= .1179*300 = 35.37. EF= .1179x300 = 35.37 .

  39. In the questions that you will solve that follow, assume N=500.If N=500, how many participants should score between the mean and a Z score of +1.30?

  40. Expected frequency = theoretical relative frequency x number of participants (EF=TRF * N). • TRF from mean to Z = +1.30 = .4032.If N = 500: EF= .4032 * 500 = 201.60

  41. Expected frequency above a score. • This is like asking the EF between your Z score and a Z score of +100.00. Half the curve lies between mu and a Z score of +100.00 • If Z is below mu, TRF is between two Z scores on opposite sides of the mean. To get TRF, add half of the curve (.5000) to the area from mu to Z. To get EF, then multiply TRF by N. • If Z is above mu, TRF is between two Z scores on the same side of the mean. To get TRF, subtract the area from mu to Z from half of the curve (.5000).To get EF, then multiply TRF by N

  42. 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 TRF above Z of -.30 is .1179 + .5000 = .6179. If N = 300: EF=.6179 x 300 = 185.37

  43. If N=500, how many participants should score above a Z score of +1.30?

  44. Expected frequency = theoretical relative frequency x number of participants (EF=TRF*N). • Z is above the mean, so you must subtract the TRF between the mean and Z from the proportion of the curve above the mean (which is half the curve or a proportion of .5000). • TRF from mu to Z = +1.30 = .4032. N = 500: EF= (.5000-.4032) * 500 = .0968*500 =48.40

  45. EF below a score. • This is the opposite of expected frequencies above a score. It is like asking the EF between your Z score and a Z score of -100.00. Half the curve lies between mu and a Z score of -100.00 • If Z is above mu, TRF is between two Z scores on opposite sides of the mean. To get TRF, add half of the curve (.5000) to the area from mu to Z. To get EF, then multiply TRF by N. • If Z is below mu, TRF is between two Z scores on the same side of the mean. To get TRF, subtract the area from mu to Z from half of the curve (.5000).To get EF, then multiply TRF by Z

  46. If N = 300, what is the EF of scores below a Z of 1.00. • Expected frequency below a score: If Z is above mu, to get TRF, add half of the curve (.5000) to the area from mu to Z.TRF below Z = +1.00 is .3413 + .5000 = .8413.

  47. Standard deviations 3 2 1 0 1 2 3 If N = 300: EF=.8413 x 300 = 252.39. F r e q u e n c y Proportion = .5000 up to mean + .3413 for 1 SD = .8413 inches

  48. Percentile Rank

  49. 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 below (to the left of) your score.

  50. 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 to Z = .4821 • Proportion at or below Z = .5000 - .4821 =.0179 • Percentile = .0179 x 100 = 1.79 = 2nd percentile

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