8: Statistical Distributions

Transcription

1 : Statistical Distributions The Uniform Distribution 1 The Normal Distribution The Student Distribution Sample Calculations The Central Limit Theory Calculations with Samples Histograms & the Normal Distribution 11 This Unit covers some of the topics of Chapters and of Quantitative Approaches in Business Studies. The reader may wish to download the files PROBABILITY.XLS, NORMALDISTA.XLS, NORMALDISTB.XLS and NORMALDISTC.XLS from the web site containing this supplement. In this Unit we will use the Data Analysis tool to generate random numbers and we will explore the behaviour of these numbers. The Uniform Distribution One of the fundamental tenets of probability theory states that experimental probability approaches theoretical probability as the number of experiments becomes very large. Most people will agree the probability of a randomly spun coin landing heads up is 0. (or 0%). This is the theoretical probability. We are not surprised if the same coin spun times gives, say, heads and tails. If we spun it 0 times we would not be surprised by heads and tails. What about a result of 0 heads and 0 tails? The theory about experimental and theoretical probabilities is difficult to prove. Whenever we get a result that disagrees with the theory we can say (a) well that can happen statistically or (b) the coin (dice, or whatever) is not random. However, the theory has stood the test of time. The worksheet on Sheet1 of PROBABILITY.XLS (See Figure 1) has 00 random numbers in column A. These were generated by using Tools Data Analysis, selecting the Random Number Generation and completing the dialog box as shown in Figure 1. We have asked for a uniform distribution of numbers in the range 1 to. By uniform we mean that every number in the range has an equal probability of being generated. In B we use the formula =INT(A) +1 and copy this down to B0 by double clicking on the cell s fill handle. The numbers in column B range from 1 to the numbers on a dice. In D:J1 we compute the experimental probability of each dice face value for various numbers of throws. The formula in E is =COUNTIF($B$:$B,E$)/COUNT($B$:$B). The numerator counts the number of times the value 1 (that is the value of E) occurs in the range B:B while the denominator counts all the values five throws. The result is the experimental probability of finding the face value 1 in throws. The dollar signs in the formula enable us to copy this formula to E:J1 and get the correct formulas in the other cells by making slight modifications. For example, in G the formula is =COUNTIF($B$:$B1,G$) / COUNT($B$:$B1). This gives the experimental probability of finding the face value in ten throws. The plot in Figure 1 shows how the experimental probabilities for two face values vary with the number of throws. The chart was made by selecting D:D1, holding down C and selecting G:G1 and J:J1. The data in D1:E1 was used to draw the horizontal line using the Edit and Paste Special method introduced in Unit under the topic Adding a New Series.

2 Statistical Distributions We can see that the probabilities for the face values and (these were chosen arbitrarily) do seem to tend to the theoretical value of 1/ or 0.1 as the number of throws increases. We may use this experiment to indicate that Excel has indeed generated a more-or-less uniform set of random numbers A B C D E F G H I J Probability Probability Random Dice Face value 1.0 n N.0 Figure 1 Probability Figure

3 Statistics Distributions The Normal Distribution The worksheet in NORMALDISTA.XLS contains three sheets that allow the reader to experiment with some properties of the normal distribution curve. Figure shows the worksheet on the TwoCurve sheet. The blue curve shows a standard normal distribution (mean = 0 and standard deviation = 1). On the y-axis we have probability values and on the x-axis we have z (measurement) values. Each point on the curve corresponds to the probability p that a measurement will yield a particular z value (value on the x-axis.) The probability is expressed as a number from 0 to 1. Of course, we could also talk about percentage probabilities just multiply p by 0. It can be shown that the area under the curve must be one since a measurement must result in some value. Note how the probability is essentially zero for any value z that is greater than standard deviations away from the mean on either side. The two parameters of the red curve may be changed by using the spinner. You will see the shape and position of the red curve alter. Just click on a spinner arrow to increase or decrease the Mean and/or StdDev. If you set the mean for the adjustable curve to zero and experiment with the standard deviation (s), you will see that as the as s increases the curve gets wider while its height decreases. The area, of course, remains constant. Figure On the second sheet (AboutM) you can select z 1 and z, one from each side of the mean, and find the probability that a measurement z will be within the range see Figure. You will see this probability written in textbooks as P(z 1 < z < z ).

4 Statistical Distributions The slider objects are used in one of three ways: (1) drag the slider bar, () for large jumps, click on the spaces either side of the slider bar, and () for more precise control, click either arrow on the slider object. Figure As shown in Figure,.% of all observations lie within two standard deviations of the mean. What are the corresponding percentages for 1 standard deviation of the mean and for - < z <? The sheet AnyP (see Figure ) is similar to the previous sheet except that the z value may take any values. To create a different visual effect, the area is plotted as a series of columns. Figure

5 The Student Distribution Statistics Distributions Figure Also in the workbook NORMALDISTA you will find the sheet Student which lets you compare the standard normal distribution with the Student distribution with varying degrees of freedom see Figure. The next Unit has some calculations using the Student distribution. Sample Calculations We will show how to perform some simple calculations involving the normal distribution. These will help the reader become familiar with the Excel function NORMDIST and its converse NORMINV. The syntax for the former is NORMDIST(x, mean, standard deviation, cumulative), where x is the measured value, mean and standard deviation have obvious meanings, and cumulative is a logical value (i.e. you may use TRUE or FALSE,or 1 or 0, for its value). A TRUE value returns the cumulative probability while a FALSE value returns the value of the probability function. There are also the functions NORMSDIST and NORMSINV which are used only for the istandard normal distribution. These, of course, do not required the mean and standard distibution arguments since the standard normal distribution these are constant at 0 and 1, respectively. Each problem will be solved in three ways: (1) using the worksheet AnyProb or AboutM, () using Appendix in Quantitative Approaches in Business Studies, and () using the NORMDIST or NORMINV function. In this way the use of the two Excel functions should become clearer. a) For a standard normal distribution, find the area under the curve to the left of z = 1.. i) The worksheet AnyProb may be used to determine the answer see Figure. You may be concerned that this computes the area from - to 1. so part of the left tail is missing. We will see that this is insignificant. This method yields.1%.

6 Statistical Distributions P A B C D E F G H z1 - Probality that Z lies between Z1 and Z z 1..1% Figure z ii) Set up a worksheet as shown in Figure and you can answer all question of this type by entering the appropriate value in B. For z = 1. we get 0.11 which agrees with the result above. A B C D E Question (a) z 1. NORMDIST 0.11 =NORMDIST(B,0,1,TRUE) Figure iii) Look up the value 1. in Appendix of Quantitative Approaches in Business Studies and you should get 0.0. Do not panic! The difference is explainable. The Appendix lists values for areas to the right as shown in the diagram in its heading. Since the total area under the curve is 1, it follows that for any given x value: Area(left of x) + Area(right of x) = 1. So the area to the left is 1-0. = 0.1. So we do get the same answer! It should be apparent by now that when we find an area to the right of x, we are finding the probability of an observation that is greater than x. Conversely, an area to the left is the probability of an observation being less than x. b) How different would our solutions be if z was negative, say -1.? The problem is now: what is the area to the left of z = -1.? i) The AnyProb worksheet gives.% or 0.0. Does that value look familiar? ii) The worksheet function NORMDIST gives the same value, i.e. 0.0 with the default format. iii) Appendix in Quantitative Approaches in Business Studies does not differentiate between positive and negative values since the normal distribution is symmetrical about the mean. So again we get 0.0. c) Find the area between z = -1. and z = 0.. i) The AnyProb worksheet gives the result.0% ii) The function NORMDIST may be used once we recognize that we need the area to the left of 0. less the area to the left of -1.. Figure shows how to set up a worksheet to solve problems of this type. So that the user does not need to remember to place the

7 Statistics Distributions two z values in order, the formula in B is =ABS(B-D). A B C D Question (c) z1 0. z -1. NORMDIST 0.0 NORMDIST 0.01 Difference 0.0 Figure iii) If you look up the two z values in Appendix the two areas are 0.1 and 0.0. The result we need is 1 - (the sum of these two), or 1 - (0.1) = 0.0. If necessary, draw a diagram to convince yourself that this is the way to proceed. Note: If you try to enter the text (c) in an Excel cell, it is most likely that the copyright symbol will be displayed. To overcome this use Tools AutoCorrect, select the appropriate entry and click the Delete button. d) Given a normal distribution with : = 00 and F = 0, what is the probability that x will have a value greater than? i) To use Appendix of Quantitative Approaches in Business Studies we must convert x µ 00 the x value to a z-score using z = or z = = 1.. Then we look up σ the z value in the Appendix to get 0.0. ii) The worksheet solution is shown in Figure. Remember that NORMDIST finds the area to the left of the x value or the probability that the observation will be less than x. We need the probability of it being greater so we use 1 - NORMDIST A B C D E F Question (d) mean 00 stdev 0 critical value P(x<value) 0.0 P(x>value) 0.0 Figure =NORMDIST(B1,B1,B1,TRUE) =1-B1 In the problems above we computed P knowing x. Sometimes we need to find x knowing P. Not surprisingly, the Excel function for this is called NORMINV and it has the syntax NORMINV(probability, mean, standard deviation). e) For a normal distribution with a mean of 0 and a standard deviation of, determine the value of x for the first %. i) The worksheet solution is shown in Figure 11 which shows an answer of A B C D E F Question (e) mean 0 stdev P % x =NORMINV(B,B1,B) Figure 11

8 Statistical Distributions ii) To solve this with the AnyProb worksheet we first find what z value will include the first %. We do this by setting z1 to - and varying z until the probability reads %, or as close to that value as we can get. We find a z value of -0. gives P =.1%. Now we must convert the z value to an x. We have already met the relationship z = x µ σ so we may write x = zσ + µ. Thus x = = 1.. This is not exactly the same as the first solution because we did not find z corresponding to exactly %. iii) To solve the problem with Appendix we use a similar hunting process. Look in the table until you find an area value close to 0.0. Did you find 0.1 with a z value of 0.? We complete the solution as in (ii) above. However, you must realise that you need the left tail of the curve so use -0. as the z value. In methods (ii) and (iii) interpolation could be used. We have these two data points P(z=0.) = 0.1 and P(z=0.) = 0.. We can say that the midpoint will be approximately P(z=0.) = 0.. which is closer to the required 0. value. With 0. as the value of z, we find x = This agrees with the worksheet approach. The Central Limit Theory The worksheet in NORMALDISTB.XLS is a demonstration of the Central Limit Theory which states : As the sample size (n = number of observations in each sample) increases, the distribution of the sampling mean ( x ) approximates a normal distribution with the mean µ = x µ and standard deviation σ σ x = / n. On sheet NormData (see Figure ) the first columns contain 0 random number taken from a normal distribution with a mean of 0 and a standard deviation of 1. In column J the averages of the nine values in each row are computed. On the sheet called CLT (Figure 1), the distribution of values in column A and in column J have been found using the FREQUENCY function and the results charted. It can be seen that as n goes from 1 to, the distribution does become approximately normal. It is left as an exercise for the reader to compute the mean and standard deviation of the data in column J of the NormData sheet and show that is is approximately 1/. 1 A B C D E F G H I J Random Normal Distribution Samples average Figure

9 Statistics Distributions A B C D E F G H I J K L M 1 sample size 1 bin frequency frequency n=1 n= Figure 1 The reader may wish to generate a new data set with the Random Number Generator found in the Data Analysis tool (see Figure 1). Figure 1 Calculations with Samples a) Coots Plc makes light bulbs whose lifetimes are distributed normally with a mean of 00 hours and a standard deviation of 0. What is the probability that a random sample of 1 bulbs for Coots will have a mean lifetime of less than hours? Solution: From the Central Limit Theory, the sampling distribution of the average ( x ) will be approximately normal with µ x = 00 and σ x = 0 / 1 = To find z corresponding 00 to x =, use z = =.. i) Again we can use the AnyProb worksheet. Strictly speaking we need the cumulative probability for the range - to -. but we will settle for - to -.. Set the spinners to these values to get an answer of 0.% or ii) Look up the absolute z in Appendix and you find the value This is the

10 Statistical Distributions probability P(z>.) but because of the symmetry of the curve it is also P(z<-.). iii) The answer may be readily found with the NORMDIST function as shown in Figure 1. The result for P(<x) in E is or 0.% A B C D E F G H I Calculations with samples. Population values Probability calculation mean 00 critical value std dev 0 P(<x) 0.% P(>x).% Sample values Size 1 Error of mean Figure 1 =B/SQRT(B) Interval calculation P % Tail size.0% x(low) 0.0 x(high) 1.0 =NORMDIST(B1,B,B,TRUE) =1-B1 =(1-E)/ =NORMINV(E11,B,B) =NORMINV(1-E11,B,B) b) If we measure a number of averages for samples of size 1, what interval around the population mean will include % of the sample means? i) If we want % distributed about the mean then there will be.% on each side. We can use the worksheet AboutM or AnyProb to find that a z value of 1.0 yields this percentage see Figure 1. Of course, on the other side of the mean, a z of -1. will encompass.%. The corresponding x values are obtained from: σ xu = µ + z n and x L = µ These give x U = 1. and x L = 0.. z σ n z1 0 z 1. Probability that Z lies between Z1 and Z.0% Figure 1 ii) Recall that Appendix gives us areas to the right of a z value the white area on the right side of the curve in Figure 1. So we need to search in the table an area value of z = 1. and we finish the problem as before. iii) A spreadsheet solution is shown in Figure 1 above where we use the NORMINV function. The same results are obtained.

11 Statistics Distributions 11 Histograms & the Normal Distribution The worksheet in the file NORMALDISTC.XLS demonstrates three ways to find the mean and the standard deviation from experimental frequency data. We will use this to add a normal curve to a histogram. Suppose you have 0 items for which it is possible to measure a quantity (weight, diameter, etc.) with one of three progressively coarser devices. On the sheet Tables (see Figure 1) the range A:B is a table listing the frequency of specific x values in a measured sample when the increment for the bins is 0.00 units. The ranges F:G and K:L1 are similar tables when the bin increments are 0.01 and 0.0, respectively. Note that the maxima for these three tables are approximately, 1 and, respectively. It is not surprising, therefore, that we shall need to normalize the curve produced by the NORMDIST function. We will assume that the distribution of these measurements are normal (i.e. Gaussian). The mean and standard deviation can be computed from such tables using the relationships: N µ = xp and σ = ( x µ ) P i i i where P i is the probability for measurement x i. N i Our data is expressed in terms of percentage frequency rather than probability but we can use the simple relationship that P i = f i /0. The mean is computed in B with the formula =SUMPRODUCT(A:A,B:B)/0. The factor of 0 comes from the P i = f i /0. The formulas in G and L are analogous: =SUMPRODUCT(F:F,G:G)/0 and =SUMPRODUCT(K:K1,L:L1)/0, respectively. To compute the standard deviation we need the value of ( xi µ ) Pi; this is the purpose of the third column in each table. The formula in C is =(A-$B$)^*B and this is copied down to row. The data in the third column is summed to give the standard deviation. So in B we use =SQRT(SUM(C:C)/0). Analogous formulas are used in the other tables. The fourth column in each table is used to compute the normal distribution values so as to be able to display a histogram with a superimposed normal curve. In D we have =NORMDIST(A,$B$,$B$,FALSE)*$D$. Carefully note the use of absolute cell references for the mean $B$, standard deviation $B$ and normalization factor $D$. This formula is copied down to row. The user may adjust the value of the normalization factor to give a total in D0 of approximately 0. There is no merit in attempting great precision here. Now we may construct a combination chart with the data from columns A, B and D. Similar methods are used with the other two tables. i i

12 Statistical Distributions A B C D E F G H I J K L M N Normal Gaussian Distribution mean 0.1 norm 0. mean 0. norm 1 mean 0.0 norm sdt 0.00 sdt 0.01 sdt 0.0 x freq (%)diff sq* prop normdist x freq (%) diff sq* prop normdist x freq (%) diff sq* prop normdist total total Figure One will find in the Excel literature two methods to fit histogram data to a normal curve that reportedly improve on the method shown above. This author has serious doubts about the supposed improvement in the results. It is doubtful if the apparent improvements in precision have any statistical significance. However, we will look at the two methods briefly. The sheet Solver1 is shown in Figure 1. The table A:B contains the same data as the middle table in the previous sheet. As before, we use the NORMDIST function to produce the normal curve. We wish to use Solver (see Unit ) to vary the three quantities mean, standard deviation and normalization factor in such a way as to make the normal curve agree with frequency data. If for a given measured value (x i ), the experimental frequency is f i and the predicted is g i, then minimizing the quantity ( fi gi) will give the so-called least-squares fit. We may compute this sum of squares of residuals (SSR) in B with the SUMXMY function as shown in Figure 1.

13 Statistics Distributions 1 A B C D E F G H I J K L Normal Gaussian Distribution mean 0. sdt 0.0 norm 1.00 SSR.01 x freq (%) normdist total 0 0. Figure 1 =SUMXMY(B:B,C:C) =NORMDIST(A,$B$,$B$,FALSE)*$B$ Copied to row Frequency (%) x Solver is set up to minimize the SSR value in B by varying the mean, standard deviation and normalization factor i.e. cells B, B and B. As a precaution, the constraint B>=0.001 is used to ensure that Solver never tries to make the standard deviation zero because the NORMDIST functions would then return error values and terminate Solver s activity. The settings for Solver are shown in Figure 1. To aid Solver in finding a solution, you may wish to start with the values found on the Tables sheet. Figure 1 A further refinement of the worksheet for use with Solver is given in Solver (see Figure 0). The formulas in column C are somewhat more complicated. This approach may give better results than the others when the increments of the x values are large. As before, Solver is used to minimize the SSR in B by varying the mean and the standard deviation.

14 1 Statistical Distributions A B C D E F G H Normal Gaussian Distribution These cells have been named l B as Mean B as std B as norm mean 0.1 std 0.0 norm 0. SSR. =NORMDIST(A,mean,std,TRUE)*norm x freq (%) normdist =(NORMDIST(A,mean,std,TRUE) NORMDIST(A,mean,std,TRUE))*norm This is copied down to row total 0 1 =(1-NORMDIST(A,mean, std,true))*norm Figure 0