six sigma statistics using excel

8/6/2019 Six Sigma Statistics Using Excel

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Statistics Using Excel Tsagris Michael

Contents

1.1 Introduction............................................................................................................4

2.1 Data Analysis..........................................................................................................5

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 ..............................................102.6 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..........................................................................18

2.11 Two-way Analysis of Variance with replication ............................................. 19

2.12 Two-way Analysis of Variance without replication........................................21

3.1 Statistical Functions.............................................................................................23

3.2 Spearmans (non-parametric) correlation coefficient ......................................27

3.3 Wilcoxon Signed Rank Test for a Median.........................................................28

3.4 Wilcoxon Signed Rank Test with Paired Data..................................................29

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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 Splus, SPSS, and Minitab e.t.c. It is reasonable not to expect that excel offersmuch 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-testfor two samples, t-test for two samples assuming (un)equal variances, paired t-

test for two samples, F-test for the equality of variances of two samples, ranksand percentiles, sampling (random and periodic, or systematic), random

numbers generation, Pearsons correlation coefficient, covariance, linear

regression, one-way ANOA, two-way ANOVA with and without replication andthe moving average. We will also demonstrate the use of non-parametric statistics inExcel for some of the previously mentioned techniques. Furthermore, informal

comparisons with the results provided by the Excel and the ones provided by SPSS

and some other packages will be carried out to see for any discrepancies between

Excel and SPSS. One thing that is worthy to mention before somebody goes throughthese notes is that they do not contain the theory underlying the techniques used.

These notes show how to cope with statistics using Excel.

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2.1 Data Analysis

If the Excel does not offer you options for statistical analyses you can add thisoption very easily. Just click on the Add-Ins option in the list ofTools. In the dialog

box (picture 2) appeared on the screen select the Analysis ToolPack. Excel will run

this command for a couple of seconds and if select Tools you will see the option Data

Analysis added on the list.

Picture 1

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Picture 2

2.2 Descriptive Statistics

The data used in most of the examples are taken from the SPSS file and refer

to car measurements (cars.sav). We just copied and pasted the data in a worksheet of

Excel. The road is always the same and mentioned already, tha is, by clicking Data

Analysis in the list of Tools. The window of picture 3 appears on the screen. We

Select Descriptive Statistics and clickOKand we are lead to the window of picture

4. In the Input Range white box we specified the data, ranging from cell 1 to cell 406all 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 value for the confidencelevel is 95%. In other words the confidence level is set to the usual 95%. The results

produced by Excel are provided under the picture 3.

Picture 3

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Picture 4

Column1

Mean 194.0418719

Standard Error 5.221297644

Median 148.5

Mode 97

Standard Deviation 105.2062324

Sample Variance 11068.35133

Kurtosis -0.79094723

Skewness 0.692125308

Range 451

Minimum 4Maximum 455

Sum 78781

Count 406

ConfidenceLevel(95.0%)

10.26422853

Table 1: Descriptive Statistics

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 notimportance. 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 thisinterval 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.

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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 fromthe 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 specific

means and variances. Due to the limited options offered by Excel we cannot test thenormality 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 samplesfrom picture 3 the window of picture 5 appears on the screen. The first column

contains the data of the first sample of size 100 while the second column is of size 80.

We selected the hypothesized mean difference to be zero and filled the white boxes ofthe variances with the specified variances. We generated two samples of sizes 100 and

80 in Splus with variances 1 and 9 respectively. The estimated variances were

different than those specified as should be. The value of the z-statistic, the critical

values and the p-values for the one-sided and two-sided tests are provided. The resultsare the same with the ones generated by Splus. 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.

Picture 5

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z-Test: Two Sample for Means

Variable 1 Variable 2

Mean 3.76501977 5.810701181

Known Variance 1 9

Observations 100 80

Hypothesized MeanDifference

0

z -5.84480403

P(Z


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t-Test: Two-Sample Assuming Unequal Variances


Mean 3.76501977 5.810701181

Variance 1.095786123 8.073733335

Observations 100 80

Hypothesized MeanDifference

0

df 96t Stat -6.115932537

P(T


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t-Test: Two-Sample Assuming Equal Variances


Mean 3.76501977 5.810701181

Variance 1.095786123 8.073733335

Observations 100 80

Pooled Variance 4.192740223

Hypothesized Mean

Difference

0

df 178

t Stat -6.660360895

P(T


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F-Test Two-Sample for Variances


Mean 3.76501977 5.8107012

Variance 1.09578612 8.0737333

Observations 100 80

df 99 79

F 0.13572236

P(F


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t-Test: Paired Two Sample for Means


Mean 3.7307128 5.81070118

Variance 1.1274599 8.07373334

Observations 80 80

Pearson Correlation 0.1785439

Hypothesized Mean

Difference

0

df 79

t Stat -6.527179

P(T


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Picture 9

If you are interested in a random sample from a know distribution then the

random numbers generation is the option you want to use. Unfortunately not manydistributions are offered. The window of this option is at picture 10. 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 distributionsoffered are Uniform, Normal, Bernoulli, Binomial, and Poisson. Two more options

are also allowed. Different distributions require different parameters to be defined.

The random seed is an option used to give the sampling algorithm a starting value butcan be left blank as well.

Picture 10

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2.9 Covariance, Correlation, Linear Regression

The covariance and correlation of two variables or two columns containing

data is very easy to calculate. The windows of correlation and covariance are the

same. We present the window of covariance.

Picture 11

Column1

Column2

Column1

1.113367

Column2

0.531949 7.972812

Table 7: Covariance

The above table is called the variance-covariance table since it produces both

of these measures. The first cell (1.113367) refers to the variance of the first column

and the last cell refers to the variance of the second column. The remaining cell(0.531949) refers to the covariance of the two columns. The blank cell is white due to

the fact that the value is the covariance (the elements of the diagonal are the variances

and the others refer to the covariance). The window of the linear regression option ispresented at picture 12. (Different normal data used in the regression analysis). 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 select the confidence interval

option. We also select the Line Fit Plots and Normal Probability Plots. Then bypressing OK, the result appears in table 8.

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Picture 12

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.875372

R Square 0.766276

Adjusted RSquare

0.76328

Standard Error 23.06123

Observations 80

ANOVA

df SS MS F Significance

FRegression 1 136001 136001 255.7274 2.46E-26

Residual 78 41481.97 531.8202

Total 79 177483

Coefficients StandardError

t Stat P-value Lower 95% Upper 95%

Intercept -10.6715 8.963642 -1.19053 0.237449 -28.5167 7.173767

X Variable 1 0.043651 0.00273 15.99148 2.46E-26 0.038217 0.049085

Table 8: Analysis of variance table

The multiple Ris the Pearson correlation coefficient, whereas the R Square

is called coefficient of determination and it is a quantity that measures the fitting ofthe model. It shows the proportion of variability of the data explained by the linear

model. The model is Y=-10.6715+0.043651*X. The adjusted R Square is thecoefficient 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. Theresults 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

0

50

100

150

200

250

0 2000 4000 6000

X Variable 1

YY

Predicted Y

Figure 1: Scatter plot of X and Y-Predicted Y

Normal Probability Plot

0

50

100

150

200

250

0 20 40 60 80 100 120

Sample Percentile

Y

Figure 2: Normal Probability Plot

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 between the two variables is obviousthrough the graph. Do not forget that the correlation coefficient exhibited a high

value. The Normal Probability Plot is used to check the normality of the residuals

graphically. Should the residuals follow the normal distribution, then the graph shouldbe 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

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normality hypothesis of the residuals. 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 theresiduals holds true, then we should see all the values within a bandwidth. We see that

almost all values fall within 40 and -40, except for two values that are over 70 and

100. These values are the so called outliers. We can assume that the residuals exhibit

constant variance. If we are not certain as for the validity of the assumption we cantransform the Y values using a log transformation and run the regression using the

transformed Y values.

-60

-40

-20

0

20

40

60

80

100

120

0 50 100 150 200 250

Predicted Values

Re

siduals

Series1

Figure 3: Residuals versus predicted values

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 thatanalysis 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 window of the one way analysis of variance is shown at picture 13.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 thewindow of picture 13, we clickOKand the results follow.

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Picture 13

Anova: Single Factor

SUMMARY

Groups Count Sum Average Variance

Column 1 253 62672 247.7154 9756.887

Column 2 73 7991 109.4658 500.5023

Column 3 79 8114 102.7089 535.4654

ANOVA

Source ofVariation

SS df MS F P-value F crit

BetweenGroups

1909939.2 2 954969.6 151.3471 1E-49 3.018168

Within Groups 2536538 402 6309.796

Total 4446477.2 404

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 affectingthe dependent variable. Now, we will see what happens when we have two factorsaffecting the dependent variable. This is called the factorial design with two factors or

two-way analysis of variance. At 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 cansee, we have three columns of data representing the three levels of the one factor and

the first columns contains only three words, S1, S2 and S3. This first column states

the three levels of the second factor. We used the S1, S2, and S3 to define the numberof the rows representing the sample sizes of each combination of the two factors. In

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other words the first combination the two factors are the cells from B2 to B26. Thismeans that each combination of factors has 24 measurements.

Picture 14

From the window of picture 3, we select Anova: Two-Factor with

replication and the window to appear is shown at picture 15.

Picture 15

We filled the two blank white boxes with the input range and Rows persample. The alpha is at its usual value, equal to 0.05. By pressing OKthe results arepresented overleaf. The results generated by SPSS are the same. At the bottom of the

table 10 there are three p-values; two p-values for the two factors and one p-value for

the interaction. The row factor is denoted as sample in Excel.

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Anova: Two-Factor With Replication

SUMMARY C1 C2 C3 Total

S1

Count 24 24 24 72

Sum 8229 2537 2378 13144

Average 342.875 105.7083 99.08333 182.5556

Variance 6668.288 237.5199 508.4275 15441.38

S2

Count 24 24 24 72

Sum 6003 2531 2461 10995

Average 250.125 105.4583 102.5417 152.7083

Variance 10582.46 416.433 515.7373 8543.364S3

Count 24 24 24 72

Sum 7629 2826 2523 12978

Average 317.875 117.75 105.125 180.25

Variance 7763.679 802.9783 664.8967 12621.15

Total

Count 72 72 72Sum 21861 7894 7362

Average 303.625 109.6389 102.25

Variance 9660.181 505.3326 553.3732

ANOVA

Source ofVariation


Sample (=Rows) 39713.18 2 19856.59 6.346116 0.002114 3.039508

Columns 1877690 2 938845.2 300.0526 6.85E-62 3.039508

Interaction 73638.1 4 18409.53 5.883638 0.000167 2.415267

Within 647689.7 207 3128.936

Total 2638731 215

Table 10: The two-way analysis of variance with replication

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 thedata as in the previous case in which the labels were necessary. We will use only the

three first three rows of the data. We still have two factors except for the fact that each

combination contains one measurement. From the window of picture 3, we select

Anova: Two-Factor without replication and the window to appear is shown atpicture 16. The only thing we did was to define the Input Range and pressed OK.

The results are presented under picture 16. What is necessary for this analysis is thatthere 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. The

disadvantage of Excel is once again that it provides no formulas for examining the

residuals in the case of analysis of variance.

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Picture 16

Anova: Two-Factor Without Replication

SUMMARY Count Sum Average Variance

Row 1 3 553 184.3333 11385.33

Row 2 3 544 181.3333 21336.33

Row 3 3 525 175 15379

Column 1 3 975 325 499

Column 2 3 340 113.3333 332.3333

Column 3 3 307 102.3333 85.33333

ANOVA

Source ofVariation


Rows 136.2222 2 68.11111 0.160534 0.856915 6.944272

Columns 94504.22 2 47252.11 111.3707 0.000311 6.944272

Error 1697.111 4 424.2778

Total 96337.56 8

Table 11: The two-way analysis of variance without replication

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3.1 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.

x AVEDEV calculates the average of the absolute deviations of the data fromtheir mean.

x AVERAGE is the mean value of all data points.

x AVERAGEA calculates the mean allowing for text values of FALSE

(evaluated as 0) and TRUE (evaluated as 1).

x BETADIST calculates the cumulative beta probability density function.

x BETAINV calculates the inverse of the cumulative beta probability densityfunction.

x BINOMDIST determines the probability that a set number of true/false trials,

where each trial has a consistent chance of 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 flips will end up heads).

x CHIDIST calculates the one-tailed probability of the chi-squared distribution.

x CHIINV calculates the inverse of the one-tailed probability of the chi-squared.Distribution.

x CHITEST calculates the result of the test for independence: the value from thechi-squared distribution for the statistics and the appropriate degrees offreedom.

x CONFIDENCE returns a value you can use to construct a confidence interval

for a population mean.

x CORREL returns the correlation coefficient between two data sets.

x COVAR calculates the covariance of two data sets. Mathematically, it is themultiplication of the correlation coefficient with the standard deviations of the

two data sets.

x CRITBINOM determines when the number of failures in a series of true/falsetrials exceeds a criterion (for example, more than 5 percent of light bulbs in

a production run fail to light).

x DEVSQ calculates the sum of squares of deviations of data points from theirsample mean. The derivation of standard deviation is very straightforward,

simply dividing by the sample size or by the sample size decreased by one to

get the unbiased estimator of the true standard deviation.

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x EXPODIST returns the exponential distribution

x FDIST calculates the F probability distribution (degree of diversity) for two

data sets.

x FINV returns the inverse of the F probability distribution.

x FISHER calculates the Fisher transformation.

x FISHERINV returns the inverse of the Fisher transformation.

x FORECAST calculates a future value along a linear trend based on an existingtime series of values.

x FREQUENCY calculates how often values occur within a range of values andthen returns a vertical array of numbers having one or more elements than

Bins_array.

x FTEST returns the result of the one-tailed test that the variances of two datasets are not significantly different.

x GAMMADIST calculates the gamma distribution.

x GAMMAINV returns the inverse of the gamma distribution.

x GAMMALN calculates the natural logarithm of the gamma distribution.

x GEOMEAN calculates the geometric mean.

x GROWTH predicts the exponential growth of a data series.

x HARMEAN calculates the harmonic mean.

x HYPGEOMDIST returns the probability of selecting an exact number of asingle type of item from a mixed set of objects. For example, a jar holds 20

marbles, 6 of which are red. If you choose three marbles, what is the

probability you will pick exactly one red marble?

x INTERCEPT calculates the point at which a line will intersect the y-axis.

x KURT calculates the kurtosis of a data set.

x LARGE returns the k-th largest value in a data set.

x LINEST generates a line that best fits a data set by generating a twodimensional array of values to describe the line.

x LOGEST generates a curve that best fits a data set by generating a two

dimensional array of values to describe the curve.

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x LOGINV returns the inverse logarithm of a value in a distribution.

x LOGNORMDIST Returns the number of standard deviations a value is awayfrom the mean in a lognormal distribution.

x MAX returns the largest value in a data set (ignore logical values and text).

x MAXA returns the largest value in a set of data (does not ignore logical valuesand text).

x MEDIAN returns the median of a data set.

x MIN returns the largest value in a data set (ignore logical values and text).

x MINA returns the largest value in a data set (does not ignore logical valuesand text).

x MODE returns the most frequently occurring values in an array or range ofdata.

x NEGBINOMDIST returns the probability that there will be a given number of

failures before a given number of successes in a binomial distribution.

x NORMDIST returns the number of standard deviations a value is away fromthe mean in a normal distribution.

x NORMINV returns a value that reflects the probability a random value

selected from a distribution will be above it in the distribution.

x NORMSDIST returns a standard normal distribution, with a mean of 0 and astandard deviation of 1.

x NORMSINV returns a value that reflects the probability a random value

selected from the standard normal distribution will be above it in thedistribution.

x PEARSON returns a value that reflects the strength of the linear relationship

between two data sets.

x PERCENTILE returns the k-th percentile of values in a range.

x PERCENTRANK returns the rank of a value in a data set as a percentage ofthe data set.

x PERMUT calculates the number of permutations for a given number ofobjects that can be selected from the total objects.

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x POISSON returns the probability of a number of events happening, given the

Poisson distribution of events.

x PROB calculates the probability that values in a range are between two limitsor equal to a lower limit.

x QUARTILE returns the quartile of a data set.

x RANK calculates the rank of a number in a list of numbers: its size relative toother values in the list.

x RSQ calculates the square of the Pearson correlation coefficient (also met as

coefficient of determination in the case of linear regression).

x SKEW returns the skewness of a data set (the degree of asymmetry of adistribution around its mean).

x SLOPE returns the slope of a line.

x SMALL returns the k-th smallest values in a data set.

x STANDARDIZE calculates the normalized values of a data set (each value

minus the mean and then divided by the standard deviation).

x STDEV estimates the standard deviation of a numerical data set based on asample of the data.

x STDEVA estimates the standard deviation of a data set (which can include

text and true/false values) based on a sample of the data.

x STDEVP calculates the standard deviation of a numerical data set.

x STDEVPA calculates the standard deviation of a data set (which can includetext and true/false values).

x STEYX returns the predicted standard error for the y value for each x value inregression.

x TDIST returns the Students t distribution

x TINV returns a tvalue based on a stated probability and degrees of freedom.

x TREND Returns values along a trend line.

x TRIMMEAN calculates the mean of a data set having excluded a percentage

of the upper and lower values.

x TTEST returns the probability associated with a Students t distribution.

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x VAR estimates the variance of a data sample.

x VARA estimates the variance of a data set (which can include text and true/

false values) based on a sample of the data.

x VARP calculates the variance of a data population.

x VARPA calculates the variance of a data population, which can include textand true/false values.

x WEIBULL calculates the Weibull distribution.

x ZTEST returns the two-tailed p-value of a z-test.

3.2 Spearmans (non-parametric) correlation

coefficientThe Spearmans correlation coefficient is the non-parametric alternative of the

Pearsons correlation coefficient. It is the Pearsons correlation coefficient based upon

the ranks of the values rather than the values. In paragraph 2.8 we exhibited how to

calculate the ranks for a range of values. The selection of calculation of the ranks willgenerate this in Excel:

Point Column1 Rank Percent

55 183 1 100.00%

43 168 2 98.50%

50 163 3 95.70%52 163 3 95.70%

63 146 5 94.30%

71 145 6 92.90%

56 141 7 90.10%

69 141 7 90.10%

1 133 9 88.70%

49 131 10 87.30%

41 130 11 85.90%

66 122 12 84.50%

6 121 13 69.00%

13 121 13 69.00%

14 121 13 69.00%22 121 13 69.00%

23 121 13 69.00%

34 121 13 69.00%

35 121 13 69.00%

47 121 13 69.00%

Table 12: Ranks and Percentiles

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Column 1 contains the values, Rankcontains the ranks of the values, Percentcontains the cumulative percentage of the values (the size of the values relative to the

others) and the first column (Points) indicates the row of each value. In the above

table, Excel has sorted the values according to their ranks. The first column indicatesthe exact position of the values. We have to sort the data with respect to this first

column, so that the format will be as in the first place. We will repeat these actions for

the second set of data and then calculate the correlation coefficient of the ranks of the

values. Attention is to be paid at the sequence of the actions described. The ranks ofthe values must be calculated separately for each data set and the sorting need to be

done before calculating the correlation coefficient. The results for the data used in this

example calculated the Spearmans correlation coefficient to be equal to 0.020483whereas the correlation calculated using SPSS is equal to 0.009. The reason for this

difference in the two correlations is that SPSS has a way of dealing the values that

have the same rank. It assigns to all values the average of the ranks. That is, if threevalues are equal (so their ranks are the same), SPSS assigns to each of these three

values the average of their ranks (Excel does not do this action).

3.3 Wilcoxon Signed Rank Test for a Median

We will now see how to conduct the Wilcoxon signed rank test for a median.

This test is based upon the ranks of the values and it is the non-parametric alternativeto the one sample t-test (when the normality assumption is not satisfied). Assume that

we are interested in testing the assumption that the median of a population from which

the sample comes from is equal to a specific median. We will use the same data set as

before. Assume that we are interested in testing whether the median is equal to 320.We calculated the median of the data set (318). This test requires some steps that must

be done carefully.

1. Step 1: Subtract all the values from the given median (i.e. 320-X i, i=1,2, , n,where n=sample size).

2. Step 2: In a new column calculate the absolute values of these subtractions.3. Step 3: Calculate the ranks of the absolute values.

4. Step 4: Using the logical function Ifdecide assign 1 if the differences in the

second column are positive and -1 if they are negative.

5. Step 5: Multiply the 4th

and the 5th

columns to get the ranks with a sign(plus/minus).

6. Step 6: Define a last column to be the squared ranks

Table 13 summarizes all of the above. All of the tedious work is complete.

Now the rest is mere details. In cases when there are values with the same ranks (i.e.ties) we use this formula for the test:

Sum(Ranks)/(Square Root of the Sum of squared ranks).

We calculate the sum of the 6th

column and of the square root of the sum of

the 7th

column. Finally, we divide the sum by the square of the second sum to get the

test statistic. In this example, the sum of squares is equal to 289, the sum of squared

ranks is equal to 117363 and its square root is equal to 342.5828. The test statistics is289 divided by 342.45828, which is equal to 0.8436. SPSS provides a little different

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test statistics due to the different handling of the tied ranks and the use of different teststatistic. There is also another way to calculate a test statistics and that is by taking the

sum of the positive ranks. Both Minitab and SPSS calculate another type of test

statistic, which is based on either the positive or the negative ranks. What is worthy tomention is that the second formula is better used in the case when there are no tied

ranks. Irrespectively of the test statistics used the result will be the same as for the

rejection of the null hypothesis. Using the second formula the result is 1401, whereas

Minitab provides a result of 1231.5. As for the result of the test (reject the nullhypothesis or not) one must look at the tables for the 1 sample Wilcoxon signed rank

test. The fact that Excel does not offer options for calculating the probabilities used in

the non-parametric tests in conjunction with the tedious work, makes it less popularfor use.

Values(Xi)

m-Xi absolute(m-Xi) Ranks of absolute values

positive ornegativedifferences

RanksRi

SquaredRanksRi

2

307 13 13 64 1 64 4096

350 -30 30 47 -1 -47 2209

318 2 2 67 1 67 4489

304 16 16 60 1 60 3600302 18 18 56 1 56 3136

429 -109 109 19 -1 -19 361

454 -134 134 13 -1 -13 169

440 -120 120 17 -1 -17 289

455 -135 135 11 -1 -11 121

390 -70 70 32 -1 -32 1024

350 -30 30 47 -1 -47 2209

351 -31 31 43 -1 -43 1849

383 -63 63 37 -1 -37 1369

360 -40 40 41 -1 -41 1681

383 -63 63 37 -1 -37 1369

Table 13: Procedure of the Wilcoxon Signed Rank Test

3.4 Wilcoxon Signed Rank Test with Paired Data

When we have two samples which cannot be assumed to be independent (i.e.

the weight of people before and after a diet) and we are interested in testing the

hypothesis that the two medians are equal versus they are not then the use of theWilcoxon signed rank test with paired data is necessary. This is the non-parametric

alternative to the paired samples t-test. The procedure is the same with the one samplecase. We will only have to add another column representing the values of the second

sample, so the table 13 would have 8 columns instead of 7 and the third columnwould be the differences between the values of the two data sets. The formulae for the

test statistics are the same as before and the results will be different from SPSS due to

the fact that Excel (in contrast to SPSS) does not manipulate ties in the ranks. Thetables of the critical values for this test must be available in order to decide whether to

reject or not the null hypothesis.

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Values Y Values X ValuesY-X

Absolutedifferences

Ranks ofabsolutedifferences

Positive ornegativedifferences

RanksRi

SquaredRanksRi

2

307 225 82 82 8 1 8 64

350 250 100 100 6 1 6 36

318 250 68 68 11 1 11 121

304 232 72 72 9 1 9 81

302 350 -48 48 13 -1 -13 169

429 400 29 29 14 1 14 196454 351 103 103 5 1 5 25

440 318 122 122 3 1 3 9

455 383 72 72 9 1 9 81

390 400 -10 10 15 -1 -15 225

350 400 -50 50 12 -1 -12 144

351 258 93 93 7 1 7 49

383 140 243 243 1 1 1 1

360 250 110 110 4 1 4 16

383 250 133 133 2 1 2 4

Table 14: Procedure of the Wilcoxon Signed Rank Test with Paired Data

six sigma statistics using excel

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