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Statistics: Purpose, Approach, Method

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The Basic Approach The basic principle behind the use of statistical tests of significance can be stated as: Compare obtained results to chance expectation. Another summation might be: Did you get what you would expect by chance?

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The Basic Approach We ask important questions: Does this obtained result differ significantly from the theoretically expected result? Does this obtained result differ from chance expectation enough to warrant a belief that something other than chance is at work? Can the obtained results be explained solely by chance? Statisticians are skeptics. They assume that results are chance results until shown to be otherwise.

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Definition and Purpose of Statistics Statistics is the theory and method of analyzing quantitative data obtained from samples of observations in order to study and compare sources of variance of phenomena, to help make decisions to accept or reject hypothesized relations between the phenomena, and to aid in drawing reliable inferences from empirical observations.

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Definition and Purpose of Statistics The first purpose is to reduce large quantities of data to manageable and understandable form. A second purpose is to aid in the study of populations and samples. A third purpose of statistics is to aid in decision making. A fourth purpose is to aid in making reliable inferences from observational data

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Definition and Purpose of Statistics To summarize much of the above discussion, the purposes of statistics can be reduced to one major purpose: to aid in inference making. Statistics says, in effect, “The inference you have drawn is correct at such-and- such a level of significance. You may act as though your hypothesis were true, remembering that there is such-and-such a probability that it is untrue.”

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Binomial Statistics Let two coins be tossed. U={ HH, HT, TH, TT}. The mean number of heads, or the expectation of heads, is M=2 ． 1/4 + 1 ． 1/4 + 1 ． 1/4 + 0 ． 1/4=1 This says that if two coins are tossed many times, the average number of heads per toss of the two coins is 1.

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Binomial Statistics In the one-toss experiment, let 1 be assigned if heads turns up and 0 if tails turn up. Then p(1)=1/2 and p(0)=1-1/2=1/2. In tossing a coin twice, let 1 be assigned to each head that occurs and 0 to be each tail. We are interested in the outcome “heads.” U={ HH, HT, TH, TT}. The mean is M=2 ． 1/4 + 1 ． 1/4 + 1 ． 1/4 + 0 ． 1/4=1 Can we arrive at the same result in an easier manner? Yes. Just add the means for each outcome. The mean of the outcome of one coin toss is ½. For two coin tosses it is ½ + ½ = 1.

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Binomial Statistics Evidently, M=p, or the mean is equal to the probability. How about a series of outcomes? In n trials the mean number of occurrences of the outcome associated with p is pn.

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The Variance V=sum[w(X)(X-M)^2] V=npq

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The Law of Large Numbers Roughly, the law says that with an increase in the size of sample, n, there is a decrease in the probability that the observed value of an event, A, will deviate from the “true” value of A by no more than a fixed amount, k. Provided the members of the samples are drawn independently, the larger the sample the closer the “true” proportion value of the population is approached.

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The Law of Large Numbers Tchebysheff’s Theorem states that if we are given a number k that is greater than or equal to 1 and a set of n measurements, we are guaranteed (regardless of the shape of the distribution) that at least (1- 1/k^2) of the measurements will lie within k standard deviation units on either side of the mean. Table 11.1

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The Normal Probability Curve and the Standard Deviation The normal probability curve is the lovely bell-shaped curve encountered so often in statistics and psychology textbooks. Its importance stems from the fact that chance events in large numbers tend to distribute themselves in the form of the curve. The so-called theory of errors uses the curve. Many phenomena—physical and psychological—are considered to distribute themselves in approximately normal form.

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The Normal Probability Curve and the Standard Deviation There are two types of graphs ordinarily used in behavioral research. One is that the values of a dependent variable are plotted against the values of an independent variable. The other is a graph that shows the distribution of a single variable. When normal distribution does not apply, use Tchebysheff’s Theorem. With this theorem, one is guaranteed 75% between Z=-2 and Z=+2 and 89.9% between Z=-3 and Z=+3.

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Interpretation of Data Using the Normal Probability Curve-Frequency Data Instead of calculating exact probabilities, we can estimate probabilities from knowledge of the properties of the normal curve. The normal curve approximation of the binomial distribution is most useful and accurate when N is large and the value of p is close to 0.5. The earlier Agree-Disagree problem can be dealt in three ways. One is chi-square test, another is the exact probability test, and the other is through normal curve.

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Interpretation of Data Using the Normal Probability Curve-Continuous Data Suppose we have the mathematics test scores of a sample of 100 fifth-grade children. The mean of the scores is 70; the standard deviation is 10. Our interest is in the reliability of the mean. How much can we depend on this mean? With future samples of similar fifth-grade children, will we get the same mean?

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Interpretation of Data Using the Normal Probability Curve-Continuous Data If we calculate a mean and a standard deviation for each of the many times, we obtain a gigantic distribution of means ( and standard deviations). The distribution will form a beautiful bell-shaped normal curve, even when the original distributions from which they are calculated are not normal. This is because we assumed “other things equal” and thus have no source of mean fluctuations other than chance.

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Interpretation of Data Using the Normal Probability Curve-Continuous Data Chance errors, given enough of them, distribute themselves into a normal distribution. This is the theory called the theory of errors.

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Interpretation of Data Using the Normal Probability Curve-Continuous Data If we had an infinite number of means from an infinite number of test administrations and calculated the mean of the means, we would then obtain the “true” mean. Naturally, we cannot do that. There is fortunately a simple way to solve the problem. It consists in accepting the mean calculated from the sample as the “true” mean and then estimating how accurate this acceptance (or assumption) is. To do this, a statistic known as the standard error of the mean is calculated. It is defined:

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Interpretation of Data Using the Normal Probability Curve-Continuous Data Means are reliable with fair-size samples. It means the standard error of mean is small enough with a larger sample to warrant the reliable means.

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Interpretation of Data Using the Normal Probability Curve-Continuous Data The standard error of the mean, then, is a standard deviation. It is a standard deviation of an infinite number of means. Only chance error makes the means fluctuate. Thus, the standard error of the means—or the standard deviation of the means, if you like—is a measure of chance or error in its effect on one measure of central tendency. A caution is in order. All the theory discussed here is based on the assumptions of random sampling and independence of observations.

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