USING EXCEL IN BIOSTATISTICS

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1 General description of Excel USING EXCEL IN BIOSTATISTICS Excel is a spreadsheet that is part ofthe Microsoft Office packages. It has other applications besides statistics. Businesses, for example, can use Excel to keep inventory records, process orders, and compare sales. Teachers could use Excel to maintain a record of student grades. It is also an excellent way to work with data in biostatistics. Some data sets are placed on the Internet in files that are readable with Microsoft ExceL Upon loading the program, it sets up a blank workbook. Each workbook is composed of sheets of information. Each workbook may contain up to 255 sheets. The sheet names are at the bottom, on tabs. You may change the name ofthe sheet by right clicking on its tab, selecting rename, and then type the new name. This is useful ifthe data is in one sheet and each graph is in a separate sheet. Each sheet is composed of columns (labeled A, B, C, etc) and rows (labeled 1,2,3, etc). Location of statistical features 1. Some functions are built-in and can be used within cells; these can be directly accessed or obtained with the Function Wizard located on the tool bar and resembling.fx. 2. Graphs are constructed with the Chart Wizard located on the tool bar and resembling a dimensional histogram. 3. Data Analysis Tool Packs are part ofexcel's custom installation. Click on Tools, Add-ins and then wait approximately 2 minutes. Then click on the top 2 data analysis tool packs. After closing the dialog box, the words Data Analysis will be included in the Tools menu. Getting started for general work Entering information into Excel. Basically, there are two major ways of first entering information; either by directly typing it in or by opening an existing file. To enter information, just type the data into cells of a worksheet. You may enter information either vertically or horizontally..you may precede each with a title, called a header row. Ifyour information is not entirely visible, you may enlarge the size of the cell by moving the cursor to the column headings (labeled A, B, C, etc.) When the cursor changes to a vertical line with arrows pointing to both the left and right, you can move the cursor to the left or right, with the mouse, thereby changing the size ofthe cell Each cell is named with its column letter followed by its row number. The name ofthe cell will appear on the left side ofthe screen, right above the worksheet. To open an existing file, click on File, Open, and then select the file to open. Usually, this will be on a disk, in drive A. You should provide your own disk for this purpose. Save the worksheet. Once the data is typed in, it is an excellent idea to save it on a disk. Click on File, Save As, and then name the file with a title that will easily identify the data. Be sure to save it on drive A. Bl

2 Manipulating the data. It may be necessary to manipulate your data into order make them easier to work with. Some of the common operations are described below. A. Applying a data format 82 Sometimes data have special formats, such as currency. To apply a special format, highlight the data to be formatted. Click on Format and select the appropriate one. 8. Inserting new data in a specific place Click on the row where the new data are to go before. Click on Insert, Row to create a new row. Likewise, a new column can be created by clicking first on the column and then Insert, Column. C. Sorting the data This is particularly useful when making a frequency table. Click on any cell within the data. Click Data, Sort and then select the fields that are to be sorted either ascending or descending. At the bottom ofthe screen, there is a place for the header row to be selected or not. D. Creating a new column, based on a calculation Suppose a frequency table has been created and it is necessary to create a relative frequency table. Assume that column A has the lower class limits and column 8 has the frequencies; both columns have header labels. In CI, type Relative Frequency. In C2, type =82 I #, where # stands for the total number ofdata items. Press Enter. The next box, C3, will be highlighted. Go back up to C2, move the cursor to the lower right corner, until it turns into a + sign. Holding down the left mouse button, pull the cursor down the column until you reach the last needed place. Excel will automatically change the formula to reflect each new location. E. Copying cells to a new location Highlight the group of cells to copy. One way to copy the cells is to press Control and C keys together. At the top ofthe new location, press Control and V keys together. F. Creating a pattern ofnumbers This is useful if when setting up class limits (for graphs) or ranks (for the normal probability plot). Type the first 2 or 3 numbers. Highlight the cells, move the mouse arrow to the lower right comer, press the le~ mouse, drag the desired amount, and release the mouse. G. Changing the width ofa column Move the pointer so that it is onthe rightmost edge ofthe column heading ofthe column to be extended. The pointer should change shape, resembling a vertical line with 2 arrows. Hold down the left mouse button and drag the width to the desired amount.

3 B3 Graphs for raw data The basic installation ofexcel has a chart wizard. Pressing a button, which looks like a three dimensional bar graph on the tool bar, accesses this. The graphs available include bar graphs, line graphs, and pie graphs. A simple default graph is highlighted in black; fancier versions are also present. To make a graph, the first thing you will need to do is to create a frequency table, using standard techniques. Ifthe table is not yet prepared, you will either need to enter the data into the calculator lists or into a column in ExceL The data must then be sorted. After the frequency table has been created, the cells must be highlighted. Press the Chart Wizard button on the top row. When moving from one screen ofthe chart wizard to the next, be sure to use the NEXT button, rather than the FINISH button. Generally, ifthe frequency table contains nominal data, Excel will not have any problems with handling the axes. If the classes are not nominal, then Excel has a slight problem. Once the chart wizard begins, you will need to click on the Series tab and remove the classes from the series to graph. On the bottom, there is a place for the category x axis. Type =(name of sheet)! (starting cell):(ending cell) for the data values. While progressing through the chart wizard, you will be allowed to enter a title for the graph and labels for both axes. The legend box could be eliminated ifdesired. The frequencies can be entered A data table could be included. The chart wizard will ask whether you to place the chart in the sheet or a chart; it is recommended to place the graph in a new chart. Click in the top circle to accomplish this. Once the graph is drawn, additional changes can be made. By clicking on the various components, you can change the font of printing, the color ofthe background, the fill pattern on the bars, the width ofthe bars, etc. For each correction, the mouse must be moved so that a little box with words describing what is to be changed appears. Double click the mouse to enter the dialog. To make the histogram bars connect, double click on a bar, go to the options tab, and run the gap width down to zero. This basic technique will work for all graphs. When preparing graphs, remember that the graph should look professional and follow correct formats. Ifa cumulative frequency polygon is desired, the graph must start with a frequency ofo. Ifa frequency polygon is desired, the graph must start and end with frequencies ofo. Be sure to include the "invisble" class marks or boundaries for these graphs. Relative frequency graphs can also be made. There are two other types ofhistogram that you may wish to make. The first one is available through a supplemental program called the Data Analysis Toolpak. This product is available separately from Duxbury Press as part ofa book called Data Analysis with Microsoft Excel. It produces a histogram with a cumulative frequency percentage graph superimposed on it. The number ofclasses (called bins) are created automatically by Excel. If it is desired to have "nice" classes, be sure to set up the lower class limits and specify the range in the bin. Be sure to check the bottom two options (Cumulative frequency and chart) to get the graph.

4 B4 This graph is placed in a sheet ofexcel, rather than in a Chart. Students will need to stretch the graph, change fonts, and generally make improvements. A sample of a modified graph is below. Average salaries of professors at 50 universities >. 15 (J c ~ 10 C" e u. 5 o +-I~ More Salary % % 80.00% 60.00%" 40.00% 20.00%..."'I % A second type ofspecial histogram is available through the Stat-Plus add-in. It is a histogram with. a nonnal curve superimposed on it, using the mean and standard deviation ofthe data. Again, be sure to specify the range ofvalues of the data. This graph is again placed in a sheet, so you will need to make changes to the design. A sample modified graph follows. Distribution of average salaries of professors at 50 universities 8 >. g 6 Q) ::::l e 4 u. 2 10,...- "--,10.00 m 8.00 ~ CD C"l ~ o t-t-t-t--i--t--i----1r-1--t '\b< '\tt- reoj r8> reo;) ro(;:j ~co ~~ '*" ~(;:j' q.' ~. rett- rere-,\(;:jo,\b<' Overall ~ 4.00 z o !!!.

5 B5 Using Excel for descriptive statistics Descriptive statistics within a cell ofa worksheet. When using any ofthe functions below, you must have a data range specified by either the name ofthe range (like Income) or the range ofthe cells (like C2:C5l). Each function is calculated by =(function name)(range name, other information needed). Calculation What it Finds Example Average Mean of the data Average(range) Trimmean Trimmed mean ofa group, the per Trimmean(range,percent) cent removed is 1/2 from the top and 1/2 from the bottom ofthe data set Median Median ofthe data Median(range) Mode Mode of the data Mode(range) *Excel only reports the first mode* Max Maximum ofthe data Max(range) Min Minimum ofthe data Min(range) Stdev Standard deviation (sample) Stdev(range) Var Variance (sample) Var(range) Stdevp Standard deviation (population) Stdevp(range) Varp Variance (population) Varp(range) Percentile Percentiles, including quartiles Percentile(range,O.##) One advantage of using this method ofobtaining descriptive statistics is that the coefficient ofvariation, range, scores within 1 or 2 standard deviations ofthe mean and scores to determine outliers can be built using previous results. For example, ifthe value ofthe mean is in cell B12 and the value ofthe standard deviation is in cell B13, then the coefficient ofvariation could be located in cell B14 by typing =100*813/812. Likewise, the limits for the number of scores within 1 standard deviation can be entered in cells B15 and B16 by typing = or = Descriptive Statistics using the Tools. Click on Tools, then select Data Analysis. A pop up menu will appear. Click on Descriptive Statistics. A selection sheet will appear. Ifthe data had been in named columns, the name ofthe column can be entered. Ifthis is the case, be sure to put a check mark in the box for labels. Otherwise, you will need to input the range ofvalues.

6 86 Near the middle ofthe box, record where you want the output moved to; a suggestion is to put it into a new sheet which needs to be named. Also, be sure to check the box labeled Descriptive Statistics. It is possible to do confidence intervals. It is also possible to print out the k th smallest and largest scores, if desired. Sample output, using a data set ofoverall faculty salaries in thousands ofdollars at 50 universities is shown below: Overall Mean Standard Error Median Mode #N/A Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 50 The #N/A for the mode tells us that there is no mode; however, ifthere is a number, be sure to check for multiple modes. Among the values given are the standard error of the mean which has the following fonnula. This forumla is part of the infonnation necessary in calculating a confidence interval The Kurtosis value is an index which describes a distribution with respect to its flatness or peakedness, as compared to the nonnal distribution. A negative value is characteristic of a relatively flat distribution while a positive value is a relatively peaked distribution. The fonnula for calculation ofkurtosis is given below. L(X-XJ4 n(n+1) 3(n-l)2 (n-l)(n-2)(n-3) s (n-2)(n-3) Another value isthat for Skewness. A negative skew indicates the longer tail extends in the direction oflow values in the distribution; the mean should be smaller than the median. A positive skew indicates the longer tail extends in the direction ofhigh values in the distribution; the mean should be larger than the median. The fonnula for skewness is x-x_j3 n (n-1xn-2) L( -s

7 B7 Box plots Box plots are a feature of Stat Plus. Select box plots to open the dialog box. Enter in the range of values for the data and check the header row label ifappropriate. Send the output to a new sheet. Moderate outliers will be shown as a filled in circle while extreme outliers will be an open circle. Extreme outliers are beyond Q3 + 3 (Q3 - Ql) or Ql- 3(Q3 - Ql). Moderate outliers are at a distance of 1.5 times the interquartile range to 3 times the interquartile range from either Ql or Q3. A sample output is shown below: Overall Scatter plots and Regression lines The Chart Wizard will also create a scatter plot ofdata. Be sure that the independent variable (x) is in the first column and the dependent variable (y) is in the second column. Highlight the data and make a the scatter plot following the general graphing procedures and storing the result in a chart. The background should be cleared. In addition, depending on the data, the axes may need to be re-scaled as Excel tends to start both at zero. To change the axes, move the mouse until you see the name of the axes appear in a little box. Double click the mouse. On the tab labeled scale, change the minimum to a value slightly smaller than the data's minimum value. To add a regression line, move the mouse near a data point. When the box with the word Series appears, right click the mouse. A drop down menu should appear. Click on Add Trendline. A series ofdifferent trendlines will appear. For this early work, the linear model is to be selected. Be sure to click the options tab and check the bottom 2 boxes - add equation and coefficient of determination to the graph. Otherwise, just the line will be drawn. The equation and value of r 2 can be moved around on the graph. Other models are available such as the exponential growth/decay model

8 B8 Binomial probabilities Excel has the binomial probability distribution built in. To begin a distribution, create 2 columns, labeled X and P(X). The X values are from 0 to n, where n is the number of trials. The Function Wizard will make it easier for the probabilities to be calculated. Put the cursor in the cell for the first P(X). Select the Function Wizard (recall it looks likeix on the toolbar). Select Statistical, then within the menu, select on Binomdist. For number, enter the cell where X = 0 was found (like A2). Enter the value for n. Enter a desired probability (the value ofp in the binomial formula). If not doing a cumulative, enter the word false. After finishing, drag the cell throughout the range of cells to compute the other binomial probabilities. Once the chart ofvalues is obtained, you could make a histogram ofthe data. Additional columns, to detennine the mean and standard deviation using the generic probability distribution formulas, could be created. Additionally, you could see the effect ofchanging the value for p on the symmetry ofthe distribution. Poisson probabilities Poisson probabilities are also available through the Function Wizard and can be utilized in much the same way. The value ofa is entered in the Wizard. Both individual and cumulative probabilities can be found. Normal Curve Probabilities Excel provides several functions related to the nonnal distribution. These are also accessed most readily through the Function Wizard which will guide you through the information to be entered. The table below describes these functions. These functions are particularly useful in establishing confidence intervals or doing hypothesis testing. Function Nonndist Nonmnv 1N0rmsdist Normsinv Standardize Accomplishes Returns the value ofthe cumulative probabilities; must have found the values for s, mean and standard deviation Returns the z score ofthe cumulative probabilities; must have cumulative percents, mean and standard deviation Returns the value ofthe cumulative probabilities; must have foudn the z score first Returns the z score ofthe cumulative probabilities; must have cumulative percents first Returns a standardized z score for a specified X, mean and standard deviation

9 B9 Central Limit Theorem To demonstrate the Central Limit Theorem, first, a population must be created. Click on Tools, Data Analysis, Random Number Generator. A dialog box will appear. The following information must be entered. Number ofvariables - to put all ofthe numbers in one column, use I Number ofrandom variables - any number that you desire - suggest 500 Distribution - the choices that would probably be appropriate are Uniform - the lower and upper bounds must be entered Normal- the mean and standard deviation must be entered Output Range - Enter the location ofthe upper left hand cell (AI, for example) Next, samples must be created - this is where the tedious, time consuming works come into play. Click on Tools, Data Analysis, Sampling. Again a dialog box will appear. The following information must be entered. Input range - the range ofvalues for the population (AI:A500, for example) Sampling method - either periodic or random; probably should use random Number ofsamples - the value for n in a ax =.j;; Output range - the first cell where the sample value is placed; values will be in a column After the first sample is made, you will need to make additional samples ofthe same size, placing results in the next column ofthe chart. This is what takes the time. However, it does show the individual samples. After all columns are created, then the sample mean ofeach column must be obtained. In a cell below the first column's data, type =Average(start cell, end cell). The formula is then copied throughout all the samples. In the example, 500 random decimals were created. Ten samples ofsize 5 were created. The output for the samples and their means are shown: Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9 Sample Means Next, at some location in the worksheet, you should find the mean and standard deviation of both the population and the sample. Also, ~ should be found, using the ~ population standard deviation and the sample size.

10 BlO Summary of values Population Mean Standard deviation Sample Mean Standard deviation Standard error What is interesting to see is that, with increased values for n, there is much less diversity in the averages. You could also graph the sample means as a histogram, ifenough samples were obtained. Normal Probability Plots One ofthe underlying assumptions that applies to all the inferential work in statistics is that the data must be normally distributed. Even though it is not usually included in texts, it is a good idea to verify this assumption, especially when working on a project where data is analyzed. Some evidence from the descriptive statistics that support (or suggest normality) are: Proximity of mean, median and mode Range is approximately 6 times the standard deviation Inter-quartile range is approximately 1.33 times the standard deviation The percent ofscores in 1 standard deviation ofthe mean is 68% The percent ofscores in 2 standard deviations ofthe mean is 95% The shape ofthe box plot The shape ofthe histogram (not recommended for small data sets) A normal probability plot, which plots the data versus the theoretical z scores ifthe data were normally distributed, should result in a straight line. Ifdata are not normally distributed, then theoretically you should attempt to normalize it by one ofseveral transformations: logarithmic, square root or reciprocal. All are readily done on ExceL The transformed data, if it appears to be a linear plot, would then be utilized for all confidence intervals and hypothesis tests, transforming final values for the interval back into the regular data values. Directionsfor a normalprobability plot: 1. Establish headings as follows: ColumnA Rank Column B Cumulative percent Column C Z score Column D Data

11 Enter into Column D, the data values. Sort the data in ascending order. Check to see if there are any duplicate data values. 3. Enter in Column A, starting in cell A2, the values from 1 to D, where n represents the number ofdata items. 4. Ifyou had no duplicate data values, skip this step and go directly to step 5. For any data points that are duplicate points, you will need to average the ranks and record that value for each ofthe duplicates. For example, ifyour worksheet has columns like below: Rank Cumulative proportion Z score Data Change it to the following: Rank Cumulative proportion Z score Data The 13's have rank 2 since the average of 1,2 and 3 is 2. The 15's have rank 5.5 since the average of5 and 6 is 5.5. rank 5. To form Column B, the cumulative proportions are found by n + I. In cell B2, enter =A2/(value for 0+1). That is, ifthe data had 30 values, enter =A2I31. Highlight the cell and drag to copy the formula down to the last row. 6. To form the inverse normal z scores (which are based on cumulative proportions and represent the area to the left ofthe z score), type in cell C2, =NORMSINV(B2). Highlight the cell and drag to copy the formula down to the last row.

12 B12 7. Use the chart wizard to create an XY scatter plot ofcolumn C versus Column D, with the Column C being the x values. After the wizard is finished, you may want ~o readjust the values on each axis~ it is not necessary for the y scale to start at zero. The plot ofthe salaries ofprofessors at 50 universities suggests that the data are close to being normally distributed. Probability Plot -I : j! z score Additionally, the Stat Plus Add-in also gives a probability plot (Pplot); this is essentially the same graph with a rotation ofaxes and a trend line added.. Normal Probability Plot ~ 0 u II) z Overall

13 B13 T distributions Excel has, through the Function Wizard, 2 functions for T distributions. These are summarized below: Function Tdist Tinv Accomplishes Returns the value ofthe probability for the tail area for a particular t score; must have the value oft, the degrees offreedom, and the number oftails (lor 2). Returns the t score for a given probability ofa tail area and degrees of freedom Confidence Intervals In real life statistics, a z interval is used only when the population standard deviation is known. Otherwise, a t interval is used. To create a z interval for raw data, the values must be entered into the worksheet. The values for the mean, standard deviation and sample size must be found and put into cells. This can be done either by the individual formulas or by the descriptive statistics add-in. For illustration, assume that the mean is in cell B2, the standard deviation is in cell B3, and the sample size is in cell B4. The steps below are the formulas to enter: Confidence level Cell B5 Enter the value ofthe confidence level as a decimal Area in tail Cell B6 =(1-B5)/2 Z score Cell B7 =ABS(NORMSINV(B6» Lower limit Cell B8 =B2-B7*B3/SQRT(B4) Upper limit Cell B9 =B2+B7*B3/SQRT(B4) Sample output is below: Z intervals Mean 54.4 Standard Deviation 4.5 Sample size 36 Confidence level 0.9 Area in tail 0.05 Z score Lower limit Upper limit To create a T interval, you have a choice. If it is desired to use formulas, then the line for Z score is replaced by: T score Cell B7 =ABS(TINV(B6,B4-1»

14 B14 Sample output is below: T intervals Mean 54.4 Standard Deviation 4.5 Sample size 36 Confidence level 0.9 Area in tail 0.05 T score Lower limit Upper limit A second method is to enter the raw data into Excel. When choosing descriptive statistics, se;. lect on the confidence level as well Sample output is below for the speeds of 10 cars: Speeds ofcars Mean 69.7 Standard Error Median 68.5 Mode #N/A Standard Deviation Sample Variance Kurtosis Skewness Range 16 Minimum 62 Maximum 78 Sum 697 Count 10 Confidence Level(95.0%) To obtain the actual interval, you must take the mean and then both subtract and add the confidence level figure (3.6794). In a similar fashion, you can create intervals for proportions. X= Cell B2 Enter the value for x N= Cell B3 Enter the value for N P estimate Cell B4 =B2/B3 Q estimate Cell B5 =1-B4 Confidence level Cell B6 Enter the value ofthe confidence level as a decimal Area in tail Cell B7 =(1-B6)/2 Z score Cell B8 =ABS(NORMSINV(B7» Lower limit Cell B9 =B3-68*SQRT(B4*B5/B3) Upper limit Cell B10 =B3+B8*SQRT(B4*B5/B3)

15 B15 Sample output is shown below: Proportion intervals x= n= p estimate q estimate confidence level area in tail z score Lower limit Upper limit Hypothesis Testing One-sample tests are not included in Excel, but as in the case ofthe intervals, formulas can be created which will accomplish the task. Sample Mean Hypothesized Mean Standard Deviation Sample size Test Statistic Alpha P value one tailed Z critical one tailed P value two tailed Z critical two tailed Cell B2 Cell B3 Cell B4 Cell B5 Cell B6 Cell B7 Cell B8 Cell B9 Cell BlO Cell B11 Enter the value here Enter the value here Enter the value here Enter the value here =(B2-B3)/(B4ISQRT(B5)) Enter the value here =1-NORMSDIST(ABS(B6» =ABS(NORMSINV(B7» =2*B8 =ABS(NORMSINV(B7/2» Sample output is shown below. One Sample Z test. Sample Mean Hypothesized Mean Standard Deviation Sample size Test statistic Alpha P value one tailed Z critical one tailed P value two tailed Z critical two tailed In a similar fashion, tests could be set up for a T test ofthe mean and a Z test for proportions. For the T test, a cell with degrees offreedom could be created. The p value formulas would require the format TDIST(location of T score, location of degrees of freedom, 1) for one tailed tests. The critical value formulas would require the format TINV(location of alpha, location of degrees of freedom) for one tailed tests; in two tailed tests, make sure the value of alpha is divided by 2.

16 B16 Fortunately, the tests for 2 populations are part ofthe Data Analysis Tool Pack, ifthe raw data are presented. Available are z tests, t tests with equal population variances, t tests with unequal population variance and paired differences t tests. For each test, you need to have the data established in columns. Select on Data Analysis, choose the test, and then fill in the desired infoonation. Note that the hypothesized mean difference is asked for; in Math 124, this is treated as zero. Sample output for paired differences is shown below: t-test: Paired Two Sample for Means Machine I Machine 2 Mean Variance Observations 7 7 Pearson Correlation Hypothesized Mean Difference o Df 6 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail ANOVA Excel has one way ANaVA as well as two way ANaVA. The data is again entered into a worksheet in either rows or columns. Enter the Data Analysis tool pack and select on one way ANaVA. Be sure that the range specified tells the location ofall of the data, including blank cells; that is, ifa box were drawn around the cells specified, all the data would be found. Sample results are shown: Anova: Single Factor Groups Count Sum Average Variance Company A CompanyB CompanyC CompanyD ANaVA Source of SS df MS F P-Value F crit Variation Between Groups Within Groups Total

17 817 Contingency Tables First, you must copy the tables into Excel (assuming that the raw data was not available). The row totals and column totals can be obtained by writing formulas and then dragging them across the table. Observed In favor Against No opinion Grand Total Men Women Grand Total To calculate the expected values, formulas for the corresponding cells need to be established. For example, the expected for Men In Favor is found by =BS*E3/ES, assuming that the worksheet was begun in cell A2. Expected In favor Against No opinion Grand Total Men Women Grand Total Next, set up headings as follows: P-value Observed chi-square Alpha Critical chi-square Click on the cell next to P-value. Click on Function Wizard, select Statistical, select CHITEST, and click Next. Enter the actual range ofobserved cell frequencies; do not include locations of totals. Below it enter the expected range of cell frequencies. Click finish. The p value of the test will appear. Click on the cell next to Observed chi-square. Click on Function Wizard, select Statistical, select CIllINV and click Next. For probability, enter the cell location of the p value. For degrees offreedom, enter the degrees of freedom which must be calculated. Click finish. The test value of Chi square will be displayed.

18 B18 Enter a value for alpha. Click on the cell next to Critical chi-square. Click on Function Wizard, select Statistical, select CIffiNV. For probability, enter the cell location ofthe p value. For degrees offreedom, enter the number. Click finish. What is really neat, is that as you change the value ofalpha, the critical value will automatically be recalculated. P-value Observed chi-square Alpha 0.01 Critical chi-square P-value Observed chi-square Alpha 0.05 Critical chi-square