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Lecture 7 THE NORMAL AND STANDARD NORMAL DISTRIBUTIONS

2. Populations and samples . When we gather data, the POPULATION is the reference set containing ALL POSSIBLE OBSERVATIONS (ALL scores, ALL reaction times, ALL IQs).Our own data are usually a selection or SAMPLE from the population. In statistics, our data are assumed to be samples from known TH

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Lecture 7 THE NORMAL AND STANDARD NORMAL DISTRIBUTIONS

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    1. 1 Lecture 7 THE NORMAL AND STANDARD NORMAL DISTRIBUTIONS

    2. 2 Populations and samples When we gather data, the POPULATION is the reference set containing ALL POSSIBLE OBSERVATIONS (ALL scores, ALL reaction times, ALL IQs). Our own data are usually a selection or SAMPLE from the population. In statistics, our data are assumed to be samples from known THEORETICAL populations.

    3. 3 Distribution of 4000 IQs

    4. 4 A sample from a population This is a picture of the distribution of 4000 IQs. The histogram is symmetrical and bell-shaped. Thats because we have sampled from a NORMAL DISTRIBUTION. The normal distribution is the most important theoretical population in statistics.

    5. 5 Large samples From the Laws of Large Numbers, we can expect the values of sample statistics to be close to those of the corresponding population parameters. The mean of this large sample is 99.9 and the SD is 14.96. These values are quite close to the population mean of 100 and the population SD of 15.

    6. 6 It makes sense to say that, if IQ is normally distributed with a mean of 100 and an SD of 15, we have sampled 4000 values from a normal distribution.

    7. 7 Population means distribution In these lectures, I shall use the terms population and distribution interchangeably.

    8. 8 Statistics versus parameters STATISTICS are characteristics of SAMPLES; PARAMETERS are characteristics of POPULATIONS. A normal population has two parameters: the mean; the standard deviation. The IQ population is (approximately) a normal distribution with a mean of 100 and an SD of 15.

    9. 9 A notational convention Letters from the Roman alphabet, such as M and s (for the mean and standard deviation, respectively), are used to denote the values of the STATISTICS of samples. Greek letters, (, s) are used to denote the values of the corresponding population characteristics or PARAMETERS.

    10. 10 The IQ example In this particular sample of 4000 IQs, M = 99.9 and s = 14.96. In the population, = 100 and s = 15.

    11. 11 The caffeine experiment In the caffeine experiment, we are sampling from TWO populations: the population of scores under the Placebo condition with mean 1 and standard deviation s1; the population of scores under the Caffeine condition with mean 2 and standard deviation s2.

    12. 12 Specifying a normal distribution Suppose that a variable X has a normal distribution with mean and standard deviation s. We write this as shown.

    13. 13 There are many normal distributions There are an infinite number of normal distributions, each specified by particular values for and s. The IQ is approximately distributed as N(100, 15). The heights of men are approximately distributed as N(69, 2.6).

    14. 14 IQs of at least 130 Suppose IQ has a normal distribution, with a mean of 100 and a standard deviation of 15. What proportion of people have IQs of at least 130?

    15. 15 At least 130? If a variable is normally distributed, 95% of values lie within 1.96 standard deviations (2 approx.) on EITHER side of the mean. So only 2 % (0.025) of values lie beyond 2 SDs above the mean.

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