Statistics using Excel 2010
Tsagris Michail
[email protected]
Athens, Nottingham and Abu Halifa 2014 1
Table of Contents 1.1 Introduction ...................................................................................................................................... 3 2.1 Data Analysis toolpack ....................................................................................................................... 4 2.2 Descriptive Statistics .......................................................................................................................... 6 2.3 Z-test for two samples ....................................................................................................................... 8 2.4 t-test for two samples assuming unequal variances ........................................................................... 9 2.5 t-test for two samples assuming equal variances ............................................................................. 10 2.6 F-test for the equality of variances ................................................................................................... 11 2.7 Paired t-test for two samples ........................................................................................................... 12 2.8 Ranks, Percentiles, Sampling, Random Numbers Generation ........................................................... 13 2.9 Covariance, Correlation, Linear Regression ...................................................................................... 15 2.10 One-way Analysis of Variance ........................................................................................................ 19 2.11 Two-way Analysis of Variance with replication ............................................................................... 20 2.12 Two-way Analysis of Variance without replication ......................................................................... 23 2.13 Statistical functions ........................................................................................................................ 24 3.1 The Solver add-in ............................................................................................................................. 27
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1.1 Introduction One of the reasons for which these notes were written was to help students and not only to perform some statistical analyses without having to use statistical software such as R, SPSS, and Minitab etc. It is reasonable not to expect that excel offers much of the options for analyses offered by statistical packages but it is in a good level nonetheless. The areas covered by these notes are: descriptive statistics , z-test for two samples , ttest for two samples assuming (un)equal variances , paired t-test for two samples , F-test for the equality of variances of two samples , ranks and percentiles , sampling (random and periodic, or systematic ), random numbers generation , Pearson’s correlation coefficient , covariance, linear regression , one-way ANOA , two-way ANOVA with and without replication and the moving average . We will also demonstrate the use of non-parametric statistics in Excel for some of the previously mentioned techniques. Furthermore, informal co mparisons with the results provided by the Excel and a nd the ones provided pro vided by SPSS and some other packages will be carried carr ied out to see for any discrepancies between Excel and SPSS. One thing that is worthy to mention before somebody goes through these notes is that they do not contain the theory underlying the techniques used. These notes not es show how to cope with statistics using Excel. The first edition was in May 2008. In the second edition (July 2012) we added the solver library. This allows us to perform linear numerical optimization (maximization/minimization) with or without linear constraints. It also offers the possibility to solve a system of equations again with or without linear constraints. I am grateful to Vassilis Vrysagotis (teaching fellow at the Technological Educational Institute of Chalkis,) for his contribution. This third edition (November 2014) uses Excel 2010 (upgrading, even in 2014). Any mistakes you find, or disagree with something stated here or anything else you want to ask, please send me an e-mail. For more statistical resources the reader is addressed to statlink.tripod.com.. statlink.tripod.com
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2.1 Data Analysis toolpack If the Excel does not o ffer you options for statistical analyses you can add t his option very easily. Just click on the File on the top left and a list will appear. From the list menu you select Options and picture 1 will appear on the screen.
Picture 1 Select add-Ins from the list on the left and the window of Picture 2 will appear. In this window (Picture 2) press Go… to move on to the window of Picture 3 where you select the two options as I did in Picture 3. If you go to the tab Data in Excel you will see the Data the Data analysis and analysis and Solver libraries added (Picture 4). The solver we will need it later. The good thing is that we only have to do this once, not every time we open the computer.
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Picture 2
Picture 3
5
Picture 4 By pressing Data pressing Data analysis (see analysis (see picture 4) the window of Picture 5 will appear.
Picture 5
2.2 Descriptive Statistics The data used in most of the examples the cars data taken from R. This data set contains information about the speed and distance covered until the automobile is stopped, of 50 cars. In the two previous versions of this document I was using the cars data (cars.sav) from SPSS. Unfortunately, I do not have these data anymore. The road will always be the same, click c lick on Data in the tools bar and from there choose Data Analysis . The dialogue box of picture 5 appears on the screen. We Select Descriptive Statistics and click OK and we are lead to the dialogue box of picture 6. In the Input Range white box we specified the data, ranging from cell 2 to cell 51 all in one column. If the first row contained label we could just define it by clicking that option. We also clicked two of the last four options (Summary statistics , Confidence Level for Mean ). As you can see the default 6
value for the confidence level is 95%. In other words the confidence level is set to the usual 95%. The results produced by Excel are provided in table 1.
Picture 6 Column1 Mean Standard Error
15.4 0.74778585
Median
15
Mode
20
Standard Deviation
5.28764444
Sample Variance
27.9591837 0.50899442 0.11750986
Kurtosis Skewness Range
21
Minimum
4
Maximum
25
Sum Count Confidence Level(95.0%)
770 50 1.50273192
Table 1: Descriptive statistics for the speed of cars. The results are pretty much the same as should be. There are only some really slight differences with regard to the rounding in the results of SPSS but of no importance. The number of observations is 406 as we expected. If there are missing values, the value in count will be less 7
than the number of rows we selected. The sample variances differ slightly but it is really not a problem. SPSS calculates a 95% confidence interval for the true mean whereas Excel provides only the quantity used to calculate the 95% confidence interval. The construction of this interval is really straightforward. Subtract this quantity from the mean to get the lower limit and add it to the mean to get the upper limit of the 95% confidence interval. So it is (mean-conf.level, mean+conf.level)=(15.4-1.50273192, 15.4+1.50273192)=(13.89727, 16.90273).
2.3 Z-test for two samples The statistical packages known to the writer do not offer the z-test for two independent samples. The results are pretty much the same with the case of the t test for two independent samples. The difference between the two tests is that apart from the normality assumption the z test assumes that we know the true variances of the two samples. We used data generated from two normal distributions with mean equal to zero for both population but different variances. Due to the limited options offered by Excel we cannot test the normality hypothesis of the data (this is also a problem met in the latter cases). Following the previously mentioned path and selecting the Z test for two samples from the dialogue box of picture 4 the dialogue box of picture 7 appears on the screen. The first column contains the data of the first sample of size 20 while the second column is of size 30. I split the speed data in two groups, the first 20 observations and the other 30.
Picture 7 We selected the hypothesized mean difference to be zero and filled the white boxes of the variances with the variances. In order to perform the z-test we must know the variance of each population from which the sample came from. S ince we do not have this information, we put the sample variances for illustration purposes. The value of the z-statistic, the critical values and the p-values for the o ne-sided and two-sided two -sided tests are provided. The T he results, provided in Table 2, are 8
the same with the ones generated by R. Both of the p-values are equal to zero, indicating that the mean difference of the two populations from which the data were drawn, is statistically significant at an alpha equal to 0.05. z-Test: Two Sample for Means Variable 1 Mean Known Variance Observations Hypothesized Mean Difference z P(Z<=z) one-tail z Critical one-tail P(Z<=z) two-tail z Critical two-tail
Variable 2
10.25
18.83333333
8.618421
11.1092
20
30
0 9.589103286 0 1.644853627 0 1.959963985
Table 2: Z-test.
2.4 t-test for two samples assuming unequal variances The theory states that when the variances of the two independent populations are not known (which is usually the case) we have to estimated them. The use of t-test is suggested in this case (but still the normality hypothesis has to be met unless the sample size is large). There are two approaches in this case; the one when we assume the variance to be equal and the one we cannot assume that. We will deal with the latter case no w. We used the same data set as before and we suppose that the variances cannot be assumed to be equal. We will see the test for the equality of two variances later. Selecting the ttest assuming unequal variances from the dialogue box of picture 4 the dialogue box of picture 8 appears on the screen. The results generated from SPSS are the same except for some rounding differences. In case you forget to set the hypothesized mean difference equal to 0, excel will use by default this number.
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Picture 8 t-Test: Two-Sample Assuming Unequal Variances Variable 1 Mean Variance
10.25
18.83333333
8.618421053
11.1091954
20
30
Observations Hypothesized Mean Difference df
Variable 2
0
t Stat
44 9.589104187
P(T<=t) one-tail
1.19886E-12
t Critical one-tail
1.680229977
P(T<=t) two-tail
2.39773E-12
t Critical two-tail
2.015367574
Table 3: t-test assuming unequal variances.
2.5 t-test for two samples assuming equal variances We will perform the same test assuming that the equality of variances ho lds true. The dialogue box for this test following following the famous path is that of picture 9. The results are the same with the ones provided by SPSS. What is worthy to mention and to pay attention is that the degrees of freedom (df ) for this case are equal to 178, whereas in the previous case were equal to 96. Also Also the t-statistics is slightly different. The reason it that different kind of formulae are used in these two cases.
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Picture 9 t-Test: Two-Sample Assuming Equal Variances Variable 1 Mean Variance Observations Pooled Variance Hypothesized Mean Difference df
Variable 2
10.25
18.83333333
8.618421053
11.1091954
20
30
10.12326389 0 48
t Stat
-9.34515099
P(T<=t) one-tail
1.10835E-12
t Critical one-tail
1.677224196
P(T<=t) two-tail t Critical two-tail
2.2167E-12 2.010634758
Table 4: t-test assuming equal variances.
2.6 F-test for the equality of variances We will now see how to test the hypothesis of the equality of variances. The dialogue box of picture 10 appears in the t he usual way by selecting the F-test from the dialogue box of picture 4. The results are the same with the ones provided by R. The p-value is equal to zero indicating that there is evidence to reject the assumption of equality of the variance of the two samples at an alpha equal with 0.05.
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Picture 10 F-Test Two-Sample for Variances Variable 1 Mean
Variable 2
10.25
18.83333333
8.618421053
11.1091954
Observations
20
30
df
19
29
Variance
F
0.775791652
P(F<=f) one-tail
0.285474981
F Critical one-tail
0.481414106
Table 5: F-test for the equa lity lity of variances.
2.7 Paired t-test for two samples Suppose that you are interested in testing the equality of two means, but the two samples (or the two populations) are not independent. For instance, when the data refer to the same people before and after a diet program. The dialogue box of picture 11 refers to this test. The results provided at table 6 are the same with the ones generated from SPSS. We can also see that the Pearson’s correlation coefficient is calculated.
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Picture 11 t-Test: Paired Two Sample for Means Variable 1 Mean Variance Observations Pearson Correlation Hypothesized Mean Difference df
Variable 2
10.25
16.95
8.618421053
3.944736842
20
20
0.941021597 0 19
t Stat
-23.76638508
P(T<=t) one-tail
6.77131E-16
t Critical one-tail
1.729132812
P(T<=t) two-tail
1.35426E-15
t Critical two-tail
2.093024054
Table 6: Paired t-test.
2.8 Ranks, Percentiles, Sampling, Random Numbers Generation The dialogue box for the Ranks and Percentiles is the one of picture 12. The use of this option is to assign a rank and a percentage at each number. The rank refers to the relative order of the number, i.e. rank 1 is assigned to the highest value; rank 2 to the second highest and so on. The percentages are of the same use.
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Picture 12 The dialogue box of the sampling option is the one in the picture 13. Two sampling schemes are available, of the systematic (periodic) and of the random sampling . In the first case case you insert a number (period), let’s say 5, means that the first value of the sample will be the number in that row (5th row) and all the rest values of the sample will be the ones of the 10th, the 15th, the 20th rows and so on. With the random sampling method, you state the sample size and Excel does the rest. If you specify a number in the second option of the sampling method, say 30, then a sample of o f size 30 will be selected from the column co lumn specified in the first box.
Picture 13 If you are interested in a random sample from a known distribution then the random numbers generation is the option you want to use. Unfortunately not many distributions are offered. The dialogue box of this option is at picture 14. In the number of variables you can select how many samples you want to be drawn from the specific distribution. The white box below is used to define the sample size. The distributions offered are Uniform, Normal , Bernoulli , Binomial , and Poisson. Two more options are also allowed. Different distributions require different parameters to be defined.
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Picture 14 The random seed is an option used to give the sampling algorithm a starting value but can be left blank as well. we ll. If we specify sp ecify a number, say 1234, then the t he next time we want to generate another sample, if we put the same random seed again we will get the same sample. The number of variables allows to generate more than one o ne samples.
2.9 Covariance, Correlation, Linear Regression The covariance and correlation of two variables or two columns containing data is very easy to calculate. The dialogue box of correlation and covariance are the same. For the correlation matrix from the dialogue box of picture 4 we select correlation.
Picture 15
15
Column 1 Column 1
1
Column 2
0.941022
Column 2 1
Table 7: Correlation matrix. The above table is called the correlation matrix. The dialogue box of the linear regression option is presented at picture 16. We fill the white boxes with the columns that represent Y and X values. The X values can contain more than one column (i.e. variable). We have to note that if one value is missing in any column the function will not be calculated. This function requires that all columns have the same number of values. Thus, if one or more columns have missing values we have to delete these rows from all columns before running the regression. We select the confidence interval option. We also select the Residual Plots , Line Fit Plots and Normal Probability Plots . The option Constant is Zero is left un-clicked. We want the constant to be in the regression line regardless of its statistical significance. By pressing OK , the result appears in table 8.
Picture 16
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SUMMARY OUTPUT Regression Statistics Multiple R
0.941021597
R Square Adjusted R Square
0.885521645 0.879161737
Standard Error
0.690416648
Observations
20
ANOVA df Regression
SS
MS
F 139.2349644
1
66.36984733
66.36984733
Residual
18
8.580152672
0.476675148
Total
19
74.95
Coefficients
Standard Error
t Stat
P-value
Significance F 6.6133E-10
Lower 95%
Upper 95%
Intercept
10.42442748
0.574168954
18.15567944
5.08222E-13
9.21814327
11.6307117
X Variable 1
0.636641221
0.053953621
11.79978663
6.6133E-10
0.523288869
0.74999357
Table 8: Analysis of variance table for the regression. The multiple R is the Pearson correlation coefficient, whereas the R Square is called coefficient of determination and it is a quantity that measures the fitting of the model. It shows the proportion of variability of the data explained by the linear model. The model is Y=10.4144+0.6366*X. The adjusted R Square is the coefficient of determination adjusted for the degrees of freedom of the model; this is a penalty of the coefficient. The p-value of the constant provides evidence to claim that the constant is not statistical significant and therefore it should be removed from the model. So, if we run the regression again we will just click on Constant is Zero , right? No, even if the constant is not significant, we still keep it in the model. Why? Because, the residuals will not sum to zero. So, do not look at the significance of the constant. Does it really matter if it’s zero or not? Does it affect the outcome? The point is to see the significance of the coefficients of the independent variables, not of the constant. Try fitting the regression line with a zero constant and check the plot of o f Figure 2 then and compare. co mpare. The results are the same generated by SPSS except for some slight differences due to roundings. The disadvantage of Excel is that it offers no normality test. The two plots also constructed by Excel are presented.
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X Variable 1 Line Fit Plot 20 15 Y
10
Y Predicted Y
5 0 0
5
10
15
X Variable 1
Figure 1: Scatter plot of X and Y & predicted Y.
X Variable 1 Residual Plot 1.2 1 s l a u d i s e R
0.8 0.6 0.4 0.2 0 0
1
2
3
4
5
X Variable 1
Figure 2: Residuals versus predicted values
The first figure is a scatter plot of the data, the X values versus the Y values and the predicted Y values. The linear relation re lation between the two variables is obvious through the graph. Do not forget that the correlati co rrelation on coefficient exhibited a high value. Excel produced also the residuals and the predicted values in the same sheet. We shall construct a scatter plot of these two values, in order to check (graphically) the assumption of homoscedasticity (i.e. constant variance through the residuals). If the assumption of heteroscedasticity of the residuals holds true, then we should see all the values within a bandwidth. We see that almost all values fall within -1.5 and 1.5. It seems like the variance is not constant since there is seems to be evidence of a pattern. This means that the residuals do not exhibit constant variance. But then again, we only have 30 points, so our eyes could be wrong. If we are not sure about the validity of the assumption we can transform the Y values using a log transformation and run the regression using the transformed Y values. 18
The Normal Probability Plot is used to check the normality of the residuals graphically. Should the residuals follow the normal distribution, then the graph should be a straight line. Unfortunately many times the eye is not the best judge of things. The Kolmogorov Smirnov test conducted in SPSS provided evidence to support the normality hypothesis of the residuals.
Normal Probability Plot 1.2 1 0.8 Y
0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
1.2
Sample Percentile
Figure 2: Normal probability plot of the residuals.
2.10 One-way Analysis of Variance The one-way analysis of variance is just the generalization of the two independent samples t-test. The assumptions the must be met in order for the results to be valid are more or less the same as in the linear regression case. It is a fact that analysis of variance and linear regression are two equivalent techniques. The Excel produces the analysis of variance table but offers no options to check the assumptions of the model. The dialogue box of the one way analysis of variance is shown at picture 17. As in the t-test cases the values of the independent variable are entered in Excel in different columns according to the factor. In our example we have three levels of the factor, therefore we have three columns. After defining the range of data in the dialogue box of picture 17, we click OK and the results follow.
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Picture 17 Anova: Single Factor SUMMARY Groups
Count
Sum
Average
Variance
Column 1
15
139
9.266667
7.495238
Column 2
12
188
15.66667
1.878788
Column 3
23
443
19.26087
15.29249
MS
F
ANOVA Source of Variation
SS
df
Between Groups
907.9652
2
453.9826
Within Groups
462.0348
47
9.830527
1370
49
Total
46.1809
P-value 8.07E12
F crit 3.195056
Table 9: The one-way analysis of variance The results generated by SPSS are very close with the results shown above. There is some difference in the sums of squares, but rather of small importance. The mean square values (MS) are very close with one another. Yet, by no means can we assume that the above results hold true since Excel does not offer options for assumptions checking.
2.11 Two-way Analysis of Variance with replication In the previous paragraph, we saw the case when we have one factor affecting the dependent variable. Now, we will see what happens when we have two factors affecting the dependent variable. This is called the factorial design with two factors or two-way analysis of variance. At 20
first, we must enter the data in the correct way. The proper way of data entry follows (the data refer to the cars measurements). As you can see, we have three columns of data representing the three levels of the one factor and the first columns contains only three words, C1, C2 and C3. This first column states the two levels of the second factor. We used the R 1, and R2 to define the number of the rows representing the sample sizes of each combination of the two factors. In other words the first combination the two factors are the cells from B2 to B6. This means that each combination of factors has 5 measurements.
Picture 18 From the dialogue box of picture 4, we select Anova: Two-Factor with replication and the dialogue box to appear is shown at picture 19.
Picture 19
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We filled the two blank white boxes with the input range and Rows per sample . The alpha is at its usual value, equal to 0.05. By pressing OK the results are presented overleaf. The results generated by SPSS are the same. At the bottom of the table 10 there are three p-values; two pvalues for the two factors and one p-value for the interaction. The row factor is denoted as sample in Excel. A limitation of this analysis when performed in Excel is that the sample sizes in each combination of column and rows (the two factors) must be equal. In other words, the design has to be balanced, the same number of values everywhere. Anova: Two-Factor With Replication SUMMARY
C1
C2
C3
Total
S1
Count
5
5
5
15
Sum
31
48
58
137
Average
6.2
9.6
11.6
9.133333
Variance
4.7
60.8
0.3
24.12381
5
5
5
15
Sum
75
130
73
278
Average
15
26
14.6
18.53333
Variance
62
34
9.3
59.98095
10
10
10
Sum
106
178
131
Average
10.6
17.8
13.1
Variance
51.15556
116.8444
6.766667
SS
df
MS
S2
Count
Total
Count
ANOVA Source of Variation Sample
F
P-value
F crit
662.7
1
662.7
23.23904
6.55E-05
4.259677
267.2667
2
133.6333
4.686148
0.019138
3.402826
Interaction
225.8
2
112.9
3.959088
0.032665
3.402826
Within
684.4
24
28.51667
1840.167
29
Columns
Total
Table 10: The table t able of the two-way analysis of variance with replication.
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2.12 Two-way Analysis of Variance without replication We will now see another case of the two-way ANOVA when each combination of factors has only one measurement. In this case we need not enter the data as in the previous case in which the labels were necessary. We will use only t he three first three rows of the data. We still have two factors except for the fact that each combination contains one measurement. From the dialogue box of picture 4, we select Anova: Two-Factor without replication and the dialogue box to appear is shown at picture 20. The only thing we did was to define the Input Range and pressed OK . The results are presented in table 11. What is necessary for this analysis is that there no interaction is present. The results are the same with the ones provided by SPSS, so we conclude once again that Excel works fine with statistical analysis. analysis. The disadvantage of Excel is once again that it provides no formulas for examining the residuals in the case of analysis of variance.
Picture 20 Anova: Two-Factor Without Replication SUMMARY
Count
Sum
Average
Variance
Row 1
3
17
5.666667
22.33333
Row 2
3
25
8.333333
14.33333
Column 1
2
8
4
0
Column 2
2
12
6
32
Column 3
2
22
11
0
ANOVA Source of Variation Rows
SS
df
MS
F
P-value
F crit
10.66667
1
10.66667
1
0.42265
18.51282
52
2
26
2.4375
0.290909
19
Error
21.33333
2
10.66667
Total
84
5
Columns
Table 11: The table of the t wo-way analysis of variance without replication. 23
2.13 Statistical functions Before showing how to find statistical measures using the statistical functions available from Excel under the Insert Function option let us see which are these.
AVEDEV calculates the average of the absolute abso lute deviations of the data from their mean. AVERAGE is the mean value of o f all data points. AVERAGEA calculates the mean allowing for text values of F ALSE (evaluated as 0) and TRUE (evaluated as 1). BETADIST calculates the cumulative beta probability density function. BETAINV calculates the inverse of the cumulative beta probability density function. BINOMDIST determines the probability that a set number of true/false trials, where each trial has a consistent chance o f generating a true or false result, will result in exactly a specified number of successes (for example, the probability that exactly four out of eight coin co in flips will end up heads). CHIDIST calculates the one-tailed probability of the chi-squared d istribution. istribution. CHIINV calculates the inverse of the one-tailed probability of the chi-squared. Distribution. CHITEST calculates the result of the test for independence: the value from the chi-square distribution for the statistics and the appropriate degrees of freedom. CONFIDENCE returns a value you can use to construct a confidence interval for a population mean. CORREL returns the correlation coefficient between two data sets. COVAR calculates the covariance of two data sets. Mathematically, Mathematically, it is the multiplication multiplication of o f the correlation coefficient with the standard deviations of the two data sets. CRITBINOM determines when the number of failures in a series of true/false trials exceeds a criterion (for example, more than 5 percent o f light bulbs in a production run fail to light). DEVSQ calculates the sum of squares of deviations o f data points from their sample mean. The derivation of standard deviat ion is very straightforward, simply simply dividing by the sample size or by the sample size decreased by one to get the unbiased estimator of the true standard deviation. EXPODIST returns the exponential distribution FDIST calculates the F probability distribution (degree o f diversity) diversity) for two data sets. FINV returns the inverse of the F probabili pro bability ty distribution. d istribution. FISHER calculates the Fisher transformation. FISHERINV returns the inverse of the Fisher transformation. FORECAST calculates a future value along a linear trend based on an existing time series of values. FREQUENCY calculates how often values occur within a range of values and then returns a vertical array of numbers having one o r more elements than Bins_array. 24
FTEST returns the result of the one-tailed one-t ailed test that the variances of two data sets are not significantly different. GAMMADIST calculates the gamma distribution. GAMMAINV returns the inverse of the gamma distribution. GAMMALN calculates the natural logarithm of the ga mma distribution. distribution. GEOMEAN calculates the geometric mean. GROWTH predicts the exponential growth of a data series. series. HARMEAN calculates the harmonic mean. HYPGEOMDIST returns the probability of selecting an exact number of a single type of item from a mixed set of o f objects. For example, a jar holds 20 marbles, 6 of which are red. If you choose choo se three marbles, what is the probability you will pick exactly one red marble? INTERCEPT calculates the point at which a line will intersect the y-axis. KURT calculates the kurtosis of a data set. LARGE returns the k-th largest value in a data set. LINEST generates a line that best fits a data set by generating a two dimensional array of values to describe the line. LOGEST generates a curve that best fits a data set by generating a two dimensional dimensional array o f values to describe the curve. LOGINV returns the inverse logarithm of a value in a d istribution. istribution. LOGNORMDIST Returns the number of standard deviations a value is away from the mean in a lognormal distribution. d istribution. MAX returns the largest value in a data set (ignore logical values and text). MAXA returns the largest value in a set of data (does not ignore logical values and text). MEDIAN returns the median of a data set. MIN returns the largest value in a data dat a set (ignore logical values and text). MINA returns the largest value in a data set (does not ignore logical values and text). MODE returns the most frequently occurring values in an array or range o f data. NEGBINOMDIST returns the probability probability that there will be a given number of failures before a given number of successes in a bino mial distribution. distribution. NORMDIST returns the number of standard deviations deviations a value is away from the mean in a normal distribution. NORMINV returns a value value that reflects the probability a random value selected from a distribution will be above it in the distribution. NORMSDIST returns a standard normal normal distribution, with a mean of 0 and a standard deviation of 1. NORMSINV returns a value value that reflects the probability a random value selected from the standard normal distribution will be above it in the distribution. PEARSON returns a value that reflects the strength of the linear relationship between two data sets. PERCENTILE returns the k-th percentile of values in a range. PERCENTRANK returns the rank of a value in a data set as a percentage of the data set. 25
PERMUT calculates the number of permutations for a given number of objects that can be selected from the tot al objects. POISSON returns the probability of a number of events happening, given the Poisson distribution of events. PROB calculates the probability that values in a range are between two limits or equal to a lower limit. QUARTILE returns the quartile of a data set. RANK calculates the rank of a number nu mber in a list of numbers: its size relative to other values in the list. RSQ calculates the square of the Pearson Pearso n correlation coefficient (also met as coefficient of determination in the case of linear regression). SKEW returns the skewness of a data set (the degree o f asymmetry of a distribution around its mean). SLOPE returns the slope of a line. SMALL returns the k-th smallest values in a data set. STANDARDIZE calculates the normalized values of a data set (each value minus the mean and then divided by the standard deviation). STDEV estimates the standard deviation of a numerical data set based on a sample of the data. STDEVA estimates the standard deviation of a data set (which can include text and true/false values) based on a sample of the data. STDEVP calculates the standard deviation of a numerical data set. STDEVPA calculates the standard deviation of a data set (which can include text and true/false values). STEYX returns the predicted standard error for the y value for ea ch x value in regression. TDIST returns the Student’s t distribution distribution TINV returns a t value based on a stated probability and degrees of freedom. TREND Returns values along a trend line. TRIMMEAN calculates the mean of a data set having excluded a percentage of the upper and lower values. TTEST returns the pro probabili bability ty associated with a Student’s t distribution. VAR estimates the variance of a data sample. VARA estimates the variance of a data set (which (which can include text and true/ false values) based on a sample of the data. VARP calculates the variance of a data population. VARPA calculates the variance of a data population, which can include text and true/false values. WEIBULL calculates the cumulative Weibull distribution. ZTEST returns the two-tailed p-value of a z -test.
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3.1 The Solver add-in Let us suppose we want to maximize the following linear bivariate function f(X,Y)=400X+300Y, under some linear constraints I. 4X+2Y ≤300 II. X≤70 III.2X+4Y≤240 The way to do it in Excel 2007 is simple. At first we will go to picture 3 and select the option Solver add-in (we already did this). Then, similarly to the data analysis path we click on Data in the tools bar and from there choose Solver. The dialogue box of picture 21 will appear. But before doing these we have to put the functions in Excel. Suppose Table 12 is the Excel. Column A
Column B
Column C
Row 1 Row 2
=A1*400+300*B1
Row 3
=4*A1+2*B1
300
Row 4
=a1
70
Row 5
=2*A1+4*B1
240
Table 12. Imitation Imitation of o f Excel. The green is the function we want to maximize. The red are t he constraints. In the cells A1 and B1 we w ill put the X and Y answers respectively. As we can see Excel offers the possibility for maximization, minimization and search for the values of X and Y which satisfy a condition, such as that the function is equal to some specific value, not only 0. Furthermore we have the option to perform the three mentioned tasks with the inclusion of constraints, either in the form of equalities or inequalities. We will use the form of inequalities in this example. exa mple.
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Picture 21 By pressing the button Add in the dialogue box of picture 21, the dialogue box of picture 22 will appear. We put the cell which describes the first constraint and the cell whose maximum value is. We repeat this task until all constraints are entered. In case we have no constraints, we do not have to come here. After the last constraint is entered we press Add. When we put the final constraint we can either press OK or press Add first and then OK . In the second case a message will appear (picture 23) preventing us from continuing. We will press Cancel and we will go to picture 24, which is the same as picture 21, but with the constraints now added.
Picture 22
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Picture 23
Picture 24 Then we select the Simplex LP as the solving method and the message of picture 25 will appear. We press OK and the message disappears. The solution will also appear in Excel.
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Picture 25 Column A Row 1
Column B 60
Column C
30
Row 2
33000
Row 3
300
300
Row 4
60
70
Row 5
240
240
Table 15: Result of the constrained co nstrained optimization. optimization.
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