• Writing About A One Way Anova Calculator
• Writing About A One Way Anova Formula
• One Way Anova Research Question

Main BodyChapter 6. F-Test and One-Way ANOVA F-distributionYears ago, statisticians discovered that when pairs of samples are taken from a normal population, the ratios of the variances of the samples in each pair will always follow the same distribution.

Not surprisingly, over the intervening years, statisticians have found that the ratio of sample variances collected in a number of different ways follow this same distribution, the F-distribution. Because we know that sampling distributions of the ratio of variances follow a known distribution, we can conduct hypothesis tests using the ratio of variances.The F-statistic is simply:where s 1 2 is the variance of sample 1. Remember that the sample variance is:Think about the shape that the F-distribution will have. If s 1 2 and s 2 2 come from samples from the same population, then if many pairs of samples were taken and F-scores computed, most of those F-scores would be close to one. All of the F-scores will be positive since variances are always positive — the numerator in the formula is the sum of squares, so it will be positive, the denominator is the sample size minus one, which will also be positive. Thinking about ratios requires some care. If s 1 2 is a lot larger than s 2 2, F can be quite large.

It is equally possible for s 2 2 to be a lot larger than s 1 2, and then F would be very close to zero. Since F goes from zero to very large, with most of the values around one, it is obviously not symmetric; there is a long tail to the right, and a steep descent to zero on the left.There are two uses of the F-distribution that will be discussed in this chapter. The first is a very simple test to see if two samples come from populations with the same variance. The second is one-way analysis of variance (ANOVA), which uses the F-distribution to test to see if three or more samples come from populations with the same mean. A simple test: Do these two samples come from populations with the same variance?Because the F-distribution is generated by drawing two samples from the same normal population, it can be used to test the hypothesis that two samples come from populations with the same variance. You would have two samples (one of size n 1 and one of size n 2) and the sample variance from each.

Obviously, if the two variances are very close to being equal the two samples could easily have come from populations with equal variances. Because the F-statistic is the ratio of two sample variances, when the two sample variances are close to equal, the F-score is close to one. If you compute the F-score, and it is close to one, you accept your hypothesis that the samples come from populations with the same variance.This is the basic method of the F-test. Hypothesize that the samples come from populations with the same variance.

Compute the F-score by finding the ratio of the sample variances. If the F-score is close to one, conclude that your hypothesis is correct and that the samples do come from populations with equal variances. If the F-score is far from one, then conclude that the populations probably have different variances.The basic method must be fleshed out with some details if you are going to use this test at work. There are two sets of details: first, formally writing hypotheses, and second, using the F-distribution tables so that you can tell if your F-score is close to one or not. Formally, two hypotheses are needed for completeness.

The first is the null hypothesis that there is no difference (hence null). It is usually denoted as H o. The second is that there is a difference, and it is called the alternative, and is denoted H 1 or H a.Using the F-tables to decide how close to one is close enough to accept the null hypothesis (truly formal statisticians would say “fail to reject the null”) is fairly tricky because the F-distribution tables are fairly tricky. Before using the tables, the researcher must decide how much chance he or she is willing to take that the null will be rejected when it is really true.

The usual choice is 5 per cent, or as statisticians say, “ α –.05″. If more or less chance is wanted, α can be varied. Choose your α and go to the F-tables. First notice that there are a number of F-tables, one for each of several different levels of α (or at least a table for each two α’s with the F-values for one α in bold type and the values for the other in regular type).

There are rows and columns on each F-table, and both are for degrees of freedom. Because two separate samples are taken to compute an F-score and the samples do not have to be the same size, there are two separate degrees of freedom — one for each sample. For each sample, the number of degrees of freedom is n-1, one less than the sample size. Going to the table, how do you decide which sample’s degrees of freedom (df) are for the row and which are for the column? While you could put either one in either place, you can save yourself a step if you put the sample with the larger variance (not necessarily the larger sample) in the numerator, and then that sample’s df determines the column and the other sample’s df determines the row. The reason that this saves you a step is that the tables only show the values of F that leave α in the right tail where F 1, the picture at the top of most F-tables shows that. Finding the critical F-value for left tails requires another step, which is outlined in the interactive Excel template in Figure 6.1.

Simply change the numerator and the denominator degrees of freedom, and the α in the right tail of the F-distribution in the yellow cells.Figure 6.1 Interactive Excel Template of an F-Table – see Appendix 6.F-tables are virtually always printed as one-tail tables, showing the critical F-value that separates the right tail from the rest of the distribution. In most statistical applications of the F-distribution, only the right tail is of interest, because most applications are testing to see if the variance from a certain source is greater than the variance from another source, so the researcher is interested in finding if the F-score is greater than one. In the test of equal variances, the researcher is interested in finding out if the F-score is close to one, so that either a large F-score or a small F-score would lead the researcher to conclude that the variances are not equal.

Because the critical F-value that separates the left tail from the rest of the distribution is not printed, and not simply the negative of the printed value, researchers often simply divide the larger sample variance by the smaller sample variance, and use the printed tables to see if the quotient is “larger than one”, effectively rigging the test into a one-tail format. For purists, and occasional instances, the left-tail critical value can be computed fairly easily.The left-tail critical value for x, y degrees of freedom (df) is simply the inverse of the right-tail (table) critical value for y, x df. Looking at an F-table, you would see that the F-value that leaves α –.05 in the right tail when there are 10, 20 df is F=2.35. To find the F-value that leaves α –.05 in the left tail with 10, 20 df, look up F=2.77 for α –.05, 20, 10 df. Divide one by 2.77, finding.36. That means that 5 per cent of the F-distribution for 10, 20 df is below the critical value of.36, and 5 per cent is above the critical value of 2.35.Putting all of this together, here is how to conduct the test to see if two samples come from populations with the same variance. First, collect two samples and compute the sample variance of each, s 1 2 and s 2 2.

Second, write your hypotheses and choose α. Third find the F-score from your samples, dividing the larger s 2 by the smaller so that F1. Fourth, go to the tables, find the table for α/2, and find the critical (table) F-score for the proper degrees of freedom ( n-1 and n-1). Compare it to the samples’ F-score.

If the samples’ F is larger than the critical F, the samples’ F is not “close to one”, and H a the population variances are not equal, is the best hypothesis. If the samples’ F is less than the critical F, H o, that the population variances are equal, should be accepted. Example #1Lin Xiang, a young banker, has moved from Saskatoon, Saskatchewan, to Winnipeg, Manitoba, where she has recently been promoted and made the manager of City Bank, a newly established bank in Winnipeg with branches across the Prairies.

After a few weeks, she has discovered that maintaining the correct number of tellers seems to be more difficult than it was when she was a branch assistant manager in Saskatoon. Some days, the lines are very long, but on other days, the tellers seem to have little to do. She wonders if the number of customers at her new branch is simply more variable than the number of customers at the branch where she used to work. Because tellers work for a whole day or half a day (morning or afternoon), she collects the following data on the number of transactions in a half day from her branch and the branch where she used to work:Winnipeg branch: 156, 278, 134, 202, 236, 198, 187, 199, 143, 165, 223Saskatoon branch: 345, 332, 309, 367, 388, 312, 355, 363, 381She hypothesizes:She decides to use α –.05. She computes the sample variances and finds:Following the rule to put the larger variance in the numerator, so that she saves a step, she finds:Figure 6.2 Interactive Excel Template for F-Test – see Appendix 6.Using the interactive Excel template in Figure 6.2 (and remembering to use the α –.025 table because the table is one-tail and the test is two-tail), she finds that the critical F for 10,8 df is 4.30.

Because her F-calculated score from Figure 6.2 is less than the critical score, she concludes that her F-score is “close to one”, and that the variance of customers in her office is the same as it was in the old office. She will need to look further to solve her staffing problem. Analysis of variance (ANOVA) The importance of ANOVAA more important use of the F-distribution is in analyzing variance to see if three or more samples come from populations with equal means.

This is an important statistical test, not so much because it is frequently used, but because it is a bridge between univariate statistics and multivariate statistics and because the strategy it uses is one that is used in many multivariate tests and procedures. One-way ANOVA: Do these three (or more) samples all come from populations with the same mean?This seems wrong — we will test a hypothesis about means by analyzing variance. It is not wrong, but rather a really clever insight that some statistician had years ago. This idea — looking at variance to find out about differences in means — is the basis for much of the multivariate statistics used by researchers today.

The ideas behind ANOVA are used when we look for relationships between two or more variables, the big reason we use multivariate statistics.Testing to see if three or more samples come from populations with the same mean can often be a sort of multivariate exercise. If the three samples came from three different factories or were subject to different treatments, we are effectively seeing if there is a difference in the results because of different factories or treatments — is there a relationship between factory (or treatment) and the outcome?Think about three samples. A group of x’s have been collected, and for some good reason (other than their x value) they can be divided into three groups. You have some x’s from group (sample) 1, some from group (sample) 2, and some from group (sample) 3. If the samples were combined, you could compute a grand mean and a total variance around that grand mean.

You could also find the mean and (sample) variance within each of the groups. Finally, you could take the three sample means, and find the variance between them. ANOVA is based on analyzing where the total variance comes from. If you picked one x, the source of its variance, its distance from the grand mean, would have two parts: (1) how far it is from the mean of its sample, and (2) how far its sample’s mean is from the grand mean.

If the three samples really do come from populations with different means, then for most of the x’s, the distance between the sample mean and the grand mean will probably be greater than the distance between the x and its group mean. When these distances are gathered together and turned into variances, you can see that if the population means are different, the variance between the sample means is likely to be greater than the variance within the samples.By this point in the book, it should not surprise you to learn that statisticians have found that if three or more samples are taken from a normal population, and the variance between the samples is divided by the variance within the samples, a sampling distribution formed by doing that over and over will have a known shape. In this case, it will be distributed like F with m-1, n– m df, where m is the number of samples and n is the size of the m samples altogether. Variance between is found by:where x j is the mean of sample j, and x is the grand mean.The numerator of the variance between is the sum of the squares of the distance between each x’s sample mean and the grand mean. It is simply a summing of one of those sources of variance across all of the observations.The variance within is found by:Double sums need to be handled with care. First (operating on the inside or second sum sign) find the mean of each sample and the sum of the squares of the distances of each x in the sample from its mean.

Second (operating on the outside sum sign), add together the results from each of the samples.The strategy for conducting a one-way analysis of variance is simple. Gather m samples. Compute the variance between the samples, the variance within the samples, and the ratio of between to within, yielding the F-score. If the F-score is less than one, or not much greater than one, the variance between the samples is no greater than the variance within the samples and the samples probably come from populations with the same mean.

If the F-score is much greater than one, the variance between is probably the source of most of the variance in the total sample, and the samples probably come from populations with different means.The details of conducting a one-way ANOVA fall into three categories: (1) writing hypotheses, (2) keeping the calculations organized, and (3) using the F-tables. The null hypothesis is that all of the population means are equal, and the alternative is that not all of the means are equal. Quite often, though two hypotheses are really needed for completeness, only H o is written:Keeping the calculations organized is important when you are finding the variance within. Remember that the variance within is found by squaring, and then summing, the distance between each observation and the mean of its sample.

Though different people do the calculations differently, I find the best way to keep it all straight is to find the sample means, find the squared distances in each of the samples, and then add those together. It is also important to keep the calculations organized in the final computing of the F-score. If you remember that the goal is to see if the variance between is large, then its easy to remember to divide variance between by variance within.Using the F-tables is the third detail. Remember that F-tables are one-tail tables and that ANOVA is a one-tail test. Though the null hypothesis is that all of the means are equal, you are testing that hypothesis by seeing if the variance between is less than or equal to the variance within.

The number of degrees of freedom is m-1, n– m, where m is the number of samples and n is the total size of all the samples together. Example #2The young bank manager in Example 1 is still struggling with finding the best way to staff her branch. She knows that she needs to have more tellers on Fridays than on other days, but she is trying to find if the need for tellers is constant across the rest of the week. She collects data for the number of transactions each day for two months.

Here are her data:Mondays: 276, 323, 298, 256, 277, 309, 312, 265, 311Tuesdays: 243, 279, 301, 285, 274, 243, 228, 298, 255Wednesdays: 288, 292, 310, 267, 243, 293, 255, 273Thursdays: 254, 279, 241, 227, 278, 276, 256, 262She tests the null hypothesis:and decides to use α –.05. She finds:m = 291.8tu = 267.3w = 277.6th = 259.1and the grand mean = 274.3She computes variance within:(276-291.8)2+(323-291.8)2(243-267.6)2(288-277.6)2(254-259.1)2/34-4=15887.6/30=529.6Then she computes variance between:9(291.8-274.3)2+9(267.3-274.3)2+8(277.6-274.3)2+8(259.1-274.3)2/4-1= 5151.8/3 = 1717.3She computes her F-score:Figure 6.3 Interactive Excel Template for One-Way ANOVA – see Appendix 6.You can enter the number of transactions each day in the yellow cells in Figure 6.3, and select the α.

As you can then see in Figure 6.3, the calculated F-value is 3.24, while the F-table (F-Critical) for α –.05 and 3, 30 df, is 2.92. Because her F-score is larger than the critical F-value, or alternatively since the p-value (0.036) is less than α –.05, she concludes that the mean number of transactions is not equal on different days of the week, or at least there is one day that is different from others.

She will want to adjust her staffing so that she has more tellers on some days than on others. SummaryThe F-distribution is the sampling distribution of the ratio of the variances of two samples drawn from a normal population. It is used directly to test to see if two samples come from populations with the same variance.

Though you will occasionally see it used to test equality of variances, the more important use is in analysis of variance (ANOVA). ANOVA, at least in its simplest form as presented in this chapter, is used to test to see if three or more samples come from populations with the same mean.

By testing to see if the variance of the observations comes more from the variation of each observation from the mean of its sample or from the variation of the means of the samples from the grand mean, ANOVA tests to see if the samples come from populations with equal means or not.ANOVA has more elegant forms that appear in later chapters. It forms the basis for regression analysis, a statistical technique that has many business applications; it is covered in later chapters. The F-tables are also used in testing hypotheses about regression results.This is also the beginning of multivariate statistics. Notice that in the one-way ANOVA, each observation is for two variables: the x variable and the group of which the observation is a part.

Writing About A One Way Anova Calculator

In later chapters, observations will have two, three, or more variables.The F-test for equality of variances is sometimes used before using the t-test for equality of means because the t-test, at least in the form presented in this text, requires that the samples come from populations with equal variances. You will see it used along with t-tests when the stakes are high or the researcher is a little compulsive.

The One-Way ANOVA ('analysis of variance') compares the means of two or more independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. One-Way ANOVA is a parametric test.This test is also known as:.

One-Factor ANOVA. One-Way Analysis of Variance. Between Subjects ANOVAThe variables used in this test are known as:. Dependent variable. Independent variable (also known as the grouping variable, or factor). This variable divides cases into two or more mutually exclusive levels, or groups.

The One-Way ANOVA is often used to analyze data from the following types of studies:. Field studies. Experiments. Quasi-experimentsThe One-Way ANOVA is commonly used to test the following:. Statistical differences among the means of two or more groups. Statistical differences among the means of two or more interventions.

Statistical differences among the means of two or more change scoresNote: Both the One-Way ANOVA and the Independent Samples t Test can compare the means for two groups. However, only the One-Way ANOVA can compare the means across three or more groups.Note: If the grouping variable has only two groups, then the results of a one-way ANOVA and the independent samples t test will be equivalent. In fact, if you run both an independent samples t test and a one-way ANOVA in this situation, you should be able to confirm that t 2= F. Your data must meet the following requirements:. Dependent variable that is continuous (i.e., interval or ratio level).

Independent variable that is categorical (i.e., two or more groups). Cases that have values on both the dependent and independent variables.

Independent samples/groups (i.e., independence of observations). There is no relationship between the subjects in each sample. The null and alternative hypotheses of one-way ANOVA can be expressed as:H 0: µ 1 = µ 2 = µ 3 =. = µ k ('all k population means are equal')H 1: At least one µ i different ('at least one of the k population means is not equal to the others')where.

µ i is the population mean of the i th group ( i = 1, 2., k)Note: The One-Way ANOVA is considered an omnibus (Latin for “all”) test because the F test indicates whether the model is significant overall—i.e., whether or not there are any significant differences in the means between any of the groups. (Stated another way, this says that at least one of the means is different from the others.) However, it does not indicate which mean is different. Determining which specific pairs of means are significantly different requires either contrasts or post hoc (Latin for “after this”) tests. The following steps reflect SPSS’s dedicated One-Way ANOVA procedure. However, since the One-Way ANOVA is also part of the General Linear Model (GLM) family of statistical tests, it can also be conducted via the Univariate GLM procedure (“univariate” refers to one dependent variable). This latter method may be beneficial if your analysis goes beyond the simple One-Way ANOVA and involves multiple independent variables, fixed and random factors, and/or weighting variables and covariates (e.g., One-Way ANCOVA).

We proceed by explaining how to run a One-Way ANOVA using SPSS’s dedicated procedure.To run a One-Way ANOVA in SPSS, click Analyze Compare Means One-Way ANOVA.The One-Way ANOVA window opens, where you will specify the variables to be used in the analysis. All of the variables in your dataset appear in the list on the left side. Move variables to the right by selecting them in the list and clicking the blue arrow buttons. You can move a variable(s) to either of two areas: Dependent List or Factor.A Dependent List: The dependent variable(s). This is the variable whose means will be compared between the samples (groups).

You may run multiple means comparisons simultaneously by selecting more than one dependent variable.B Factor: The independent variable. The categories (or groups) of the independent variable will define which samples will be compared. The independent variable must have at least two categories (groups), but usually has three or more groups when used in a One-Way ANOVA.C Contrasts: (Optional) Specify contrasts, or planned comparisons, to be conducted after the overall ANOVA test.When the initial F test indicates that significant differences exist between group means, contrasts are useful for determining which specific means are significantly different when you have specific hypotheses that you wish to test. Contrasts are decided before analyzing the data (i.e., a priori). Contrasts break down the variance into component parts. They may involve using weights, non-orthogonal comparisons, standard contrasts, and polynomial contrasts (trend analysis).Many online and print resources detail the distinctions among these options and will help users select appropriate contrasts. Please see the IBM SPSS guide for detailed information on Contrasts by clicking the?

Button at the bottom of the dialog box.D Post Hoc: (Optional) Request post hoc (also known as multiple comparisons) tests. Specific post hoc tests can be selected by checking the associated boxes.1 Equal Variances Assumed: Multiple comparisons options that assume homogeneity of variance (each group has equal variance). For detailed information about the specific comparison methods, click the Help button in this window.2 Test: By default, a 2-sided hypothesis test is selected. Alternatively, a directional, one-sided hypothesis test can be specified if you choose to use a Dunnett post hoc test.

Click the box next to Dunnett and then specify whether the Control Category is the Last or First group, numerically, of your grouping variable. In the Test area, click either Control. The one-tailed options require that you specify whether you predict that the mean for the specified control group will be less than ( Control) or greater than (. To introduce one-way ANOVA, let's use an example with a relatively obvious conclusion.

The goal here is to show the thought process behind a one-way ANOVA. Problem StatementIn the sample dataset, the variable Sprint is the respondent's time (in seconds) to sprint a given distance, and Smoking is an indicator about whether or not the respondent smokes (0 = Nonsmoker, 1 = Past smoker, 2 = Current smoker).

Let's use ANOVA to test if there is a statistically significant difference in sprint time with respect to smoking status. Sprint time will serve as the dependent variable, and smoking status will act as the independent variable. Before the TestJust like we did with the paired t test and the independent samples t test, we'll want to look at descriptive statistics and graphs to get picture of the data before we run any inferential statistics.The sprint times are a continuous measure of time to sprint a given distance in seconds. From the Descriptives procedure ( Analyze Descriptive Statistics Descriptives), we see that the times exhibit a range of 4.5 to 9.6 seconds, with a mean of 6.6 seconds (based on n=374 valid cases). From the Compare Means procedure ( Analyze Compare Means Means), we see these statistics with respect to the groups of interest:NMeanStd.

DeviationNonsmoker2616.4111.252Past smoker336.8351.024Current smoker597.1211.084Total3536.5691.234Notice that, according to the Compare Means procedure, the valid sample size is actually n=353. This is because Compare Means (and additionally, the one-way ANOVA procedure itself) requires there to be nonmissing values for both the sprint time and the smoking indicator.Lastly, we'll also want to look at a comparative boxplot to get an idea of the distribution of the data with respect to the groups:From the boxplots, we see that there are no outliers; that the distributions are roughly symmetric; and that the center of the distributions don't appear to be hugely different.

Writing About A One Way Anova Formula

The median sprint time for the nonsmokers is slightly faster than the median sprint time of the past and current smokers. Running the Procedure. Click Analyze Compare Means One-Way ANOVA. Add the variable Sprint to the Dependent List box, and add the variable Smoking to the Factor box.

Click Options. Check the box for Means plot, then click Continue.

One Way Anova Research Question

Click OK when finished.Output for the analysis will display in the Output Viewer window. Syntax ONEWAY Sprint BY Smoking/PLOT MEANS/MISSING ANALYSIS. OutputThe output displays a table entitled ANOVA.Sum of SquaresdfMean SquareFSig.Between Groups26.7.209.000Within Groups5.455Total535.870352After any table output, the Means plot is displayed.The Means plot is a visual representation of what we saw in the Compare Means output.

The points on the chart are the average of each group. It's much easier to see from this graph that the current smokers had the slowest mean sprint time, while the nonsmokers had the fastest mean sprint time. Discussion and ConclusionsWe conclude that the mean sprint time is significantly different for at least one of the smoking groups ( F 2, 350 = 9.209, p.