Customizable tables of p values
for teaching the t, F, and chi-square distributions

Unpublished manuscript
by Paul T. von Hippel

Abstract

Many textbooks still use critical values to teach hypothesis tests, and use statistical tables that were designed with critical values in mind. We present statistical tables that allow students to look up p values directly, eliminating the need to teach the critical-value approach. The tables are generated using statistical functions in Microsoft Excel spreadsheets; by editing the spreadsheets, instructors can modify the tables to suit their tastes.

Key words: hypothesis test, critical value, significance level, p value

Many textbooks still teach a cumbersome and old-fashioned approach to hypothesis testing, in which the observed test statistic is compared to the critical value associated with a fixed significance level. Most applied researchers, however, simply report the p value, or indidate whether the p value is below the significance level.

Students find p values easily when analyzing data on a computer, but they have more difficulty when working problems by hand. The difficulty comes from the fact that many textbooks use old-fashioned statistical tables designed for the critical-value approach. In principle, textbook tables can be used to look up p values. In practice, though, p values obtained in this way can be rather coarse, since textbook tables often give just a handful of probabilities.

Table 1 summarizes the dimensions of t tables in a convenient sample of 27 textbooks and references published in the past decade or so. (This updates the summary in Dawson 1997.) Teachers who favor p values will find some books more hospitable than others. A teacher using Ritchey (2000), for example, may find it difficult to teach p values, since the t table in that book gives only 3 different probabilities. A teacher using Walpole., Myers, Myers, & Ye (2002), however, will have an easier time, since that book's t table gives 14 different probabilities -- though none of these exceeds .40.

Tables 2 and 3 summarize the chi-square and F tables in the same books. Chi-square tables often give 10 or more probabilities, but F tables rarely give more than 4 traditional significance levels (e.g., .10, .05, .025, .01). Using textbook F tables, it is difficult to avoid discussing critical values, even if the topic has been successfully avoided in teaching the t and chi-square.

Table 1. Dimensions of t tables

Book	Probabilities	df values	Pages
Ritchey (2000)	3	34	1
Thorne & Giessen (2000)	4	42	1
Agresti & Finlay (1997)	5	30	1
Freund (2004)	5	30	1
Hogg & Craig (1995)	5	30	1
Bartz (1999)	6	38	2
Freedman, Pisani, & Purves (1998)	6	25	1
Levin & Fox (2004)	6	34	1
Frankfort-Nachmias & Leon-Guerrerro (2002)	6	34	2
Gravetter & Wallnau (2004)	6	34	1
Kendrick (2005)	6	34	2
Triola (2001)	6	30	1
Zwillinger & Kokoska (2000)	7	36	1
Sincich (1993)	7	34	1
Siegel & Morgan (1996)	7	35	2.5
Rice (1995)	8	34	1
Lunneborg (1994)	8	45	2
Rossman, Chance, & Lock (2001)	8	43	2
Dean & Voss (1999)	9	43	1
Kirk (1995)	9	34	1
Johnson & Wichern (1999)	9	34	1
Milton & Arnold (2003)	9	101	2
Spiegel & Liu (1999)	10	34	1
Moore & McCabe (2006)	12	37	1
Watkins, Scheaffer, & Cobb (2004)	12	37	1
Walpole, Myers, Myers, & Ye (2002)	14	34	2
Neter et al. (1996)	14	34	2
Minimum	3	25	1
Median	7	34	1
Maximum	14	101	2.5

Table 2. Dimensions of chi-square tables

Book	Probabilities	df values	Pages
Thorne & Giessen (2000)	2	30	0.5
Levin & Fox (2004)	2	30	1
Bartz (1999)	3	30	0.5
Ritchey (2000)	4	34	1
Siegel & Morgan (1996)	4	100	4
Gravetter & Wallnau (2004)	5	37	1
Lunneborg (1994)	5	44	2
Hogg & Craig (1995)	6	30	1
Agresti & Finlay (1997)	7	29	1
Freund (2004)	8	30	1
Freedman, Pisani, & Purves (1998)	9	20	1
Johnson & Wichern (1999)	9	37	1
Neter et al. (1996)	9	37	1
Rossman, Chance, & Lock (2001)	9	35	2
Rice (1995)	10	23	1
Triola (2001)	10	37	1
Dean & Voss (1999)	10	41	1
Sincich (1993)	10	42	2
Watkins, Scheaffer, & Cobb (2004)	11	35	1
Moore & McCabe (2006)	12	35	1
Spiegel & Liu (1999)	13	37	1
Milton & Arnold (2003)	13	30	2
Kirk (1995)	14	30	1
Frankfort-Nachmias & Leon-Guerrerro (2002)	14	30	1
Kendrick (2005)	14	30	2
Zwillinger & Kokoska (2000)	16	82	4
Walpole, Myers, Myers, & Ye (2002)	20	30	2
Minimum	2	20	0.5
Median	9	34	1
Maximum	20	100	4

Table 3. Dimensions of F tables

Book	Probabilities	df values		Pages
		Numerator	Denominator
Levin & Fox (2004)	2	8	34	2
Ritchey (2000)	2	8	34	2
Thorne & Giessen (2000)	2	9	33	2
Kendrick (2005)	2	10	34	2
Frankfort-Nachmias & Leon-Guerrerro (2002)	2	11	34	2
Spiegel & Liu (1999)	2	16	40	2
Freund (2004)	2	19	30	2
Bartz (1999)	2	9	33	2.5
Gravetter & Wallnau (2004)	2	15	52	3
Walpole, Myers, Myers, & Ye (2002)	2	19	34	4
Lunneborg (1994)	2	19	34	4
Milton & Arnold (2003)	2	43	43	11
Hogg & Craig (1995)	3	12	12	2
Agresti & Finlay (1997)	3	11	34	3
Johnson & Wichern (1999)	3	17	34	6
Dean & Voss (1999)	3	19	30	6
Triola (2001)	3	19	35	6
Siegel & Morgan (1996)	4	8	24	4
Rice (1995)	4	19	34	4
Kirk (1995)	4	30	24	6
Sincich (1993)	4	19	34	8
Moore & McCabe (2006)	5	20	36	8
Zwillinger & Kokoska (2000)	6	13	25	6
Neter et al. (1996)	7	18	18	6
Minimum	2	8	12	2
Median	2.5	16.5	34	4
Maximum	7	43	52	11

Note. There are no F tables in the textbooks by Freedman, Pisani, & Purves (1998), Rossman, Chance, & Lock (2001), or Watkins, Scheaffer, & Cobb (2004).

At first glance, increasing the number of tabled p values would seem to require substantially larger tables (e.g., Piegorsch 2002). This seems especially impractical for the F distribution, where the typical layout requires one or two pages for every significance level.

But in fact the space in statistical tables can be used more efficiently than it usually is. An increase in the number of p values can be balanced by a decrease in the number of degrees of freedom (df), since most statistical tables give more df values than are necessary for practical purposes.

Textbook t tables give between 25 and 101 different df values, typically including all of the df values between 1 and 25. As Dawson (1997) pointed out, many of these df values are redundant since, even in the tails, there are only trivial differences among the t distributions with (say) 18-22 degrees of freedom.
Textbook chi-square tables give at least 20 df values, often including values of df=100 or higher. While none of these values is redundant, some may be unnecessary. In many introductory textbooks, the chi-square statistic is only used to test homogeneity of one- or two-way frequency tables. In such a course, students will never need values for df=100 since they will never work hand problems involving an 11-by-11 table. (Some instructors may use chi-square for different purposes, such as estimating variance components. For those instructors, high df values may seem necessary. On the other hand, chi-square problems with high df can often be worked using a normal approximation.)
Most F tables give more than 16 values for the between-groups degrees of freedom dfb, sometimes including values of dfb=100 or higher. Again, most of these values are unnecessary in an introductory course that uses the F test for basic one-way ANOVA (or simple regression). Unless you have students hand-analyze experiments with 17 treatment levels (or hand-solve regressions with 16 predictors), they will never need to look up F values for dfb=16 -- much less for dfb=100.
Most F tables also give at least 34 values for the within-groups degrees of freedom dfw, including all of the dfw values between 1 and 25. As in the t table, many of these dfw values are unnecessary since there are only trivial differences among the F distributions having (say) 18-22 degrees of freedom within groups.

Capitalizing on these observations, this paper lays out expanded tables of p values using Microsoft Excel.

The t table below is adapted from one proposed by Dawson (1997). The table gives 21 different p values, which is somewhat fewer than Dawson's 33, but still 50% more than any textbook. Like Dawson's t table, the table below makes room for the extra p values by giving just 19 df values, where textbooks give at least 25. As remarked by Dawson, the omitted df values provide little useful information.

Table 4. The t distribution. For Excel source, click here.

Although the basic design of this t table comes from Dawson, I have made alterations to suit my tastes. Since the modified table uses Excel, other instructors can make further changes easily (see also Hunt 1997). For example:

To add a row for a two-tailed p value of .25, simply highlight the row with p=.20; choose Rows from the Insert menu; then type .25 in the appropriate cell of the newly created row (cell U17). The rest of the row fills automatically.
Likewise, to add a column for 75 degrees of freedom, highlight the column with df=50; choose Columns from the Insert menu; and type 75 in the appropriate cell of the newly created column (cell D12). To fill in the new column, highlight the t values for df=50 and drag left (or choose Fill->Left from the Edit menu). The correct t values appear automatically.

Values for t are calculated using Excel's TINV function. This function loses accuracy as p approaches zero, but it is accurate, to the displayed number of digits, for the smallest p values in the table (Knüsel 1998; McCullough & Wilson 2002).

Following similar designs, Table 5 displays a chi-square table with 1 to 25 degrees of freedom, and Table 6 displays an F table with 1 to 8 degrees of freedom between groups. The chi-square table is not so different from the one in Walpole et al. (1998), but the F table gives three times as many p values as any of the sampled textbooks.

Table 5. The chi-square distribution. For Excel source, click here.

Table 6. The F distribution with dfb=1 degree of freedom between groups. F values for dfb=2 appear further down the same worksheet, and F values for dfb=3 to 8 appear on later worksheets. For Excel source, click here.

The chi-square and F tables appear in the same Excel file as the t table. To move from one table to another in Excel, click the tabs along the bottom of the spreadsheet (labeled t, chi, F(1,2), etc.).

As remarked above, the F and chi-square tables have more than enough degrees of freedom for the hand problems that are assigned in many introductory statistics courses. Problems with more df would typically be worked on a computer. However, instructors who need more degrees of freedom will find the tables easy to modify. For example, to create F tables with dfb=9 and 10, simply highlight and copy the contents from one of the F worksheets (e.g., F(7,8)); create a new worksheet by choosing Worksheet from the Insert menu; paste the copied material into the new worksheet; and edit the values in the cells labeled dfb. The appropriate F values appear automatically.

F and chi-square values are calculated using the Excel functions FINV and CHIINV. These functions are accurate for the tabled p values, though they would be less accurate for p values very close to zero. The CHIINV function is also inaccurate for p values very close to 1, but it is accurate for the largest p value in the table (p=.99).

All of the tables can be printed at once by selecting Print from the File menu, and choosing to print the Entire workbook. The t and chi-square tables print on one side of paper each, while the F tables print on four sides of paper. I include the printed tables in my course binder, and use them instead of tables from a textbook.

I have been teaching with these tables for about a year. They have made it much simpler to teach hypothesis tests. I no longer talk about critical values, and I rarely mention significance levels. Students simply look up the p value, interpet it, and decide whether the evidence against the null hypothesis is convincing. I teach them that it is conventional to reject the null hypothesis if p<.05 (or p<.01), but I point out that these thresholds are arbitrary and that it seems silly to make much fuss over the difference between (say) p=.04 and p=.06. This point is easy to illustrate using a table that actually reports values for p=.04 and p=.06.

Using these tables, I no longer waste time addressing the confusions associated with critical values. I no longer have to belabor the distinction between the observed value and the critical value, or between the significance level and the p value. This frees up time to talk about more important issues, such as the interpetation of the p value, the relationship between confidence intervals and hypothesis tests, and the distinction between statistical significance and substantive effect size.

References

Agresti, A., & Finlay, B. (1997), Statistical Methods for the Social Sciences (3rd ed.), Upper Saddle River, NJ: Prentice-Hall.

Bartz, A.E. (1999), Basic Statistical Concepts (4th ed.), Upper Saddle River, NJ: Prentice-Hall.

Dawson, R.J.M. (1997), Turning the Tables: A t-Table for Today. Journal of Statistics Education 5(2).

Dean, A.., & Voss, D. (1998), Design and Analysis of Experiments, New York: Springer.

Frankfort-Nachmias, C., & Leon-Guerrerro, A. (2002), Social Statistics for a Diverse Society(3rd ed.), Thousand Oaks, CA: Pine Forge.

Freedman, D., Pisani, R., & Purves, R. (1998), Statistics (3rd ed.), New York: Norton.

Freund, J.E. (2004), Modern Elementary Statistics (11th ed.), Upper Saddle River, NJ: Prentice Hall.

Gravetter, F.J., & Wallnau, L.B. (2004), Statistics for the Behavioral Sciences (6th ed.), Belmont, CA: Wadsworth.

Hogg, R.V., & Craig, A.T. (1995), Introduction to Mathematical Statistics. Upper Saddle River, NJ: Prentice Hall.

Hunt, N. (1997), What price statistical tables now? Teaching Statistics, 19(2), 49-51.

Johnson, R.A. & Wichern, D.W. (1999), Applied Multivariate Statistical Methods (4th ed.), Upper Saddle River, NJ: Prentice-Hall.

Kirk, R.E. (1995), Experimental Design: Procedures for the Behavioral Sciences (3rd ed.), Pacific Grove, CA: Brooks/Cole.

Kendrick, J.R. (2005), Social Statistics: An Introduction Using SPSS for Windows (2nd ed.), Boston: Pearson/AB Longman.

Knüsel, L. (1998), On the accuracy of statistical distributions in Microsoft Excel 97. Computational Statistics and Data Analysis 26, 375-377.

Levin, J., & Fox, J.A. (2000), Elementary Statistics in Social Research: The Essentials. Boston: Pearson/AB Longman.

McCullough, B.D., & Wilson, B. (2002), On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP. Computational Statistics & Data Analysis 40, 713-721.

Milton, J.S., & Arnold, J.C. (2003), Introduction to Probability and Statistics (4th ed.), Boston: McGraw-Hill.

Moore, D. S., & McCabe, G. P. (2006), Introduction to the Practice of Statistics (5th ed.), New York: Freeman.

Neter, J., Kutner, M.H., Nachtsheim, C.J., & Wasserman, W. (1996), Applied Linear Regression Models (3rd ed.). Chicago: Irwin.

Piegorsch, W.W. (2002). Tables of p values for t and chi-square reference distributions. University of South Carolina Statistics Technical Report No. 194. Available at http://www.stat.sc.edu/~piegorsc/TR194.pdf.

Rice, J.A. (1995), Mathematical Statistics and Data Analysis, Belmont, CA: Duxbury/Wadsworth.

Ritchey, F. (2000), The Statistical Imagination, Boston: McGraw-Hill.

Rossman, A.J., Chance, B.L., & Lock. R.H. (2001), Workshop Statistics: Discovery with Data and Fathom. Emeryville, CA: Key Curriculum Press.

Siegel, A.F., & Morgan, C.J. (1996). Statistics and Data Analysis (2nd ed.), New York: Wiley.

Sincich, T. (1993), Statistics by Example (5th ed.), New York: MacMillan.

Spiegel, M.R. & Liu, J. (1999), Mathematical Handbook of Formulas and Tables, New York: McGraw-Hill (Schaum's Outline Series).

Thorne, B.M., & Giessen, J.M. Statistics for the Behavioral Sciences (3rd ed.), Mountain View, CA: Mayfield.

Triola, M.F. (2001), Elementary Statistics (8th ed.), Boston: Addison-Wesley.

Walpole, R. E., Myers, R. H., Myers, S.L., & Ye, K. (2002), Probability and Statistics for Engineers and Scientists (7th ed.), New York: Macmillan.

Watkins, A.E., Scheaffer, R.L., & Cobb, G.W. (2004), Statistics in Action: Understanding a World of Data. Emeryville, CA: Key Curriculum Press.

Zwillinger, D. & Kokoska, S. (2000), CRC Standard Probability and Statistics Tables and Formulae (6th ed.), Boca Raton, FL: Chapman & Hall / CRC.

Customizable tables of p values for teaching the t, F, and chi-square distributions

Abstract

References

Customizable tables of p values
for teaching the t, F, and chi-square distributions