1 Normal Distributions Heibatollah Baghi, and Mastee Badii

2 The Normal Curve A mathematical model or and an idealized conception of the form a distribution might have taken under certain circumstances. –Mean of any distribution has a Normal distribution (Central Limit Theorem) –Many observations (height of adults, weight of children in California, intelligence) have Normal distributions Shape –Bell shaped graph, most of data in middle –Symmetric, with mean, median and mode at same point

3 Percent of Values Within One Standard Deviations 68.26% of Cases

4 Percent of Values Within Two Standard Deviations 95.44% of Cases

5 Percent of Values Within Three Standard Deviations 99.72% of Cases

6 Percent of Values Greater than 1 Standard Deviation

7 Percent of Values Greater than -2 Standard Deviations

8 Percent of Values Greater than +2 Standard Deviations

9 Data in Normal Distribution

10 Properties Of Normal Curve Normal curves are symmetrical. Normal curves are unimodal. Normal curves have a bell-shaped form. Mean, median, and mode all have the same value.

11 Standard Scores One use of the normal curve is to explore Standard Scores. Standard Scores are expressed in standard deviation units, making it much easier to compare variables measured on different scales. There are many kinds of Standard Scores. The most common standard score is the ‘z’ scores. A ‘z’ score states the number of standard deviations by which the original score lies above or below the mean of a normal curve.

12 The Z Score The normal curve is not a single curve but a family of curves, each of which is determined by its mean and standard deviation. In order to work with a variety of normal curves, we cannot have a table for every possible combination of means and standard deviations.

13 The Z Score What we need is a standardized normal curve which can be used for any normally distributed variable. Such a curve is called the Standard Normal Curve.

14 The Standard Normal Curve The Standard Normal Curve (z distribution) is the distribution of normally distributed standard scores with mean equal to zero and a standard deviation of one. A z score is nothing more than a figure, which represents how many standard deviation units a raw score is away from the mean.

15 Example Z Score For scores above the mean, the z score has a positive sign. Example + 1.5z. Below the mean, the z score has a minus sign. Example - 0.5z. Calculate Z score for blood pressure of 140 if the sample mean is 110 and the standard deviation is 10 Z = 140 – 110 / 10 = 3

16 Comparing Scores from Different Distributions Interpreting a raw score requires additional information about the entire distribution. In most situations, we need some idea about the mean score and an indication of how much the scores vary. For example, assume that an individual took two tests in reading and mathematics. The reading score was 32 and mathematics was 48. Is it correct to say that performance in mathematics was better than in reading?

17 Z Scores Help in Comparisons Not without additional information. One method to interpret the raw score is to transform it to a z score. The advantage of the z score transformation is that it takes into account both the mean value and the variability in a set of raw scores.

18 Did Sara improve? Score in pretest was 18 and post test was 42 Sara’s score did increase. From 18 to 42. But her relative position in the Class decreased. PretestPost test Observation1842 Mean1749 Standard deviation349 Z score

19 Area When Score is Known For a normal distribution with mean of 100 and standard deviation of 20, what proportion of cases fall below 80? ~16%

20 Score When Area Is Known For a normal distribution with mean of 100 and standard deviation of 20, find the score that separates the upper 20% of the cases from the lower 80% Answer = 116.8

21 Transforming Standard Scores Sometimes it is more convenient to work with standard scores that do not have negative numbers or decimals. Standard scores can be transformed to have any desired mean and standard deviation. SAT and GRE are transformed scores (similar to z) with a mean of 500 and an SD of 100 – (score x 100) Widely used cognitive and personality test (Wechsler IQ test) are standardized to have a mean of 100 and an SD of 15 – ( z x 15) + 100

22 Transforming a raw score of 12 on Behavioral Problem Index Age 5: Mean: 10.0 SD: 2.0 Age 6: Mean: 12.0SD: 3.0 Age 7: Mean: 14.0SD: 3.0

23 Transforming a raw score of 12 on Behavioral Problem Index Age 5: Mean: 10.0 SD: 2.0 Age 6: Mean: 12.0SD: 3.0 Age 7: Mean: 14.0SD: 3.0 Age 5: Z = (12-10) / 2 = 1.0 Age 6: Z = (12-12) / 3 = 0.0 Age 7:Z = (12-14) / 3 = -0.67

24 Transforming a raw score of 12 on Behavioral Problem Index Age 5: Mean: 10.0 SD: 2.0 Age 6: Mean: 12.0SD: 3.0 Age 7: Mean: 14.0SD: 3.0 Age 5: Z = (12-10) / 2 = 1.0 Age 6: Z = (12-12) / 3 = 0.0 Age 7:Z = (12-14) / 3 = Age 5: Standard Score =(1.0 X 15) + 100= 115 Age 6: Standard Score =(0.0 X 15) + 100= 100 Age 7: Standard Score =(-0.67 X 15) +100= 90

25 Other Standard Scores A T score is created from a z score simply by multiplying each standard deviation unit by 10 to get rid of the decimals, and then adding 50 to each of these scores to get rid of the negatives. Now the mean becomes 50 ([10*0] + 50 = 50). Plus 1 z becomes 60 ([10*1] + 50 = 60).

26 Multiple Transformation of Data

27 The Normal Curve & Probability The normal curve also is central to many aspects of inferential statistics. This is because the normal curve can be used to answer questions concerning the probability of events. For example, by knowing that 16% of adults have a Wechsler IQ greater than 115 (z = +1.00), one can state the probability(p) of randomly selecting from the adult population a person whose IQ is greater than 115. You are correct if you suspect that P is.16.

28 Data on the IQ Scores of 1000 Six Grade Children

29 The Normal Curve & Probability The mean of the distribution is 100 and the SD is 15 What is the probability that a randomly selected student from this population would have an IQ score of 115 or greater? Approximately percent of the total area under the curve in the distribution