Interpret all statistics and graphs for One-Way ANOVA

Find definitions and interpretation guidance for every statistic and graph that is provided with one-way ANOVA.

Adj MS

Adjusted mean squares measure how much variation a term or a model explains, assuming that all other terms are in the model, regardless of the order they were entered. Unlike the adjusted sums of squares, the adjusted mean squares consider the degrees of freedom.

The adjusted mean square of the error (also called MSE or s2) is the variance around the fitted values.

Interpretation

Minitab uses the adjusted mean squares to calculate the p-value for a term. Minitab also uses the adjusted mean squares to calculate the adjusted R2 statistic. Usually, you interpret the p-values and the adjusted R2 statistic instead of the adjusted mean squares.

Adjusted p-value

The adjusted p-value indicates which pairs within a family of comparisons are significantly different. The adjustment limits the family error rate to the alpha level that you specify. If you use a regular p-value for multiple comparisons, the family error rate increases with each additional comparison.

It is important to consider the family error rate when making multiple comparisons, because your chances of committing a type I error for a series of comparisons is greater than the error rate for any one comparison alone.

Interpretation

If the adjusted p-value is less than alpha, reject the null hypothesis and conclude that the difference between a pair of group means is statistically significant. The adjusted p-value also represents the smallest family error rate at which a particular null hypothesis is rejected.

Adj SS

Adjusted sums of squares are measures of variation for different components of the model. The order of the predictors in the model does not affect the calculation of the adjusted sum of squares. In the Analysis of Variance table, Minitab separates the sums of squares into different components that describe the variation due to different sources.

Adj SS Term
The adjusted sum of squares for a term is the increase in the regression sum of squares compared to a model with only the other terms. It quantifies the amount of variation in the response data that is explained by each term in the model.
Adj SS Error
The error sum of squares is the sum of the squared residuals. It quantifies the variation in the data that the predictors do not explain.
Adj SS Total
The total sum of squares is the sum of the term sum of squares and the error sum of squares. It quantifies the total variation in the data.

Interpretation

Minitab uses the adjusted sums of squares to calculate the p-value for a term. Minitab also uses the sums of squares to calculate the R2 statistic. Usually, you interpret the p-values and the R2 statistic instead of the sums of squares.

Boxplot

A boxplot provides a graphical summary of the distribution of each sample. The boxplot makes it easy to compare the shape, the central tendency, and the variability of the samples.

Interpretation

Use a boxplot to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than 20.

Skewed data

Examine the spread of your data to determine whether your data appear to be skewed. When data are skewed, the majority of the data are located on the high or low side of the graph. Skewed data indicates that the data might not be normally distributed. Often, skewness is easiest to detect with an individual value plot, a histogram, or a boxplot.

Right-skewed
Left-skewed

The boxplot with right-skewed data shows average wait times. Most of the wait times are relatively short, and only a few of the wait times are longer. The boxplot with left-skewed data shows failure rate data. A few items fail immediately, and many more items fail later.

Data that are severely skewed can affect the validity of the p-value if your sample is small (< 20 values). If your data are severely skewed and you have a small sample, consider increasing your sample size.

Outliers

Outliers, which are data values that are far away from other data values, can strongly affect your results. Often, outliers are easiest to identify on a boxplot.

On a boxplot, asterisks (*) denote outliers.

Try to identify the cause of any outliers. Correct any data-entry errors or measurement errors. Consider removing data values for abnormal, one-time events (special causes). Then, repeat the analysis.

Confidence Interval for group means (95% CI)

These confidence intervals (CI) are ranges of values that are likely to contain the true mean of each population. The confidence intervals are calculated using the pooled standard deviation.

Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeat your sample many times, a certain percentage of the resulting confidence intervals contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.

The confidence interval is composed of the following two parts:
Point estimate
The point estimate is the parameter that is calculated from the sample data. The confidence interval is centered around this value.
Margin of error
The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the error margin is added to the point estimate. To calculate the lower limit of the confidence interval, the error margin is subtracted from the point estimate.

Interpretation

Use the confidence interval to assess the estimate of the population mean for each group.

For example, with a 95% confidence level, you can be 95% confident that the confidence interval contains the group mean. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size.

Means
Paint
N
Mean
StDev
Pooled StDev = 3.95012

In these results, each blend has a confidence interval for its mean hardness. The multiple comparison results for these data show that Blend 4 is significantly harder than Blend 2. That Blend 4 is harder than Blend 2 does not show that Blend 4 is hard enough for the intended use of the paint. The confidence interval for the group mean is better for judging whether Blend 4 is hard enough.

DF

The total degrees of freedom (DF) are the amount of information in your data. The analysis uses that information to estimate the values of unknown population parameters. The total DF is determined by the number of observations in your sample. The DF for a term show how much information that term uses. Increasing your sample size provides more information about the population, which increases the total DF. Increasing the number of terms in your model uses more information, which decreases the DF available to estimate the variability of the parameter estimates.

If two conditions are met, then Minitab partitions the DF for error. The first condition is that there must be terms you can fit with the data that are not included in the current model. For example, if you have a continuous predictor with 3 or more distinct values, you can estimate a quadratic term for that predictor. If the model does not include the quadratic term, then a term that the data can fit is not included in the model and this condition is met.

The second condition is that the data contain replicates. Replicates are observations where each predictor has the same value. For example, if you have 3 observations where pressure is 5 and temperature is 25, then those 3 observations are replicates.

If the two conditions are met, then the two parts of the DF for error are lack-of-fit and pure error. The DF for lack-of-fit allow a test of whether the model form is adequate. The lack-of-fit test uses the degrees of freedom for lack-of-fit. The more DF for pure error, the greater the power of the lack-of-fit test.

Difference of Means

This value is the difference between the sample means of two groups.

Interpretation

The differences between the sample means of the groups are estimates of the differences between the populations of these groups.

Because each mean difference is based on data from a sample and not from the entire population, you cannot be certain that it equals the population difference. To better understand the differences between population means, use the confidence intervals.

Equal variances

Minitab assumes that the population standard deviations for all groups are equal.

Interpretation

Look in the standard deviation (StDev) column of the one-way ANOVA output to determine whether the standard deviations are approximately equal.

If you cannot assume equal variances, use Welch's ANOVA, which is an option for one-way ANOVA that is available in Minitab Statistical Software.

Fisher Individual Tests for Differences of Means

Use the individual confidence intervals to identify statistically significant differences between the group means, to determine likely ranges for the differences, and to determine whether the differences are practically significant. Fisher's individual tests table displays a set of confidence intervals for the difference between pairs of means.

The individual confidence level is the percentage of times that a single confidence interval includes the true difference between one pair of group means, if you repeat the study. Individual confidence intervals are available only for Fisher's method. All of the other comparison methods produce simultaneous confidence intervals.

Controlling the individual confidence level is uncommon because it does not control the simultaneous confidence level, which often increases to unacceptable levels. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

The confidence interval of the difference is composed of the following two parts:

Point estimate
The point estimate is the difference between a pair of means and is calculated from the sample data. The confidence interval is centered on this value.
Error margin
The error margin defines the width of the confidence interval and is determined by the observed variability in the sample and the confidence level. To calculate the upper limit of the confidence interval, the error margin is added to the point estimate. To calculate the lower limit of the confidence interval, the error margin is subtracted from the point estimate.

Interpretation

Use the confidence intervals to assess the differences between group means.

Fisher Individual Tests for Differences of Means
Difference of Levels
Difference of Means
SE of Difference
T-Value
Adjusted P-Value
Simultaneous confidence level = 80.83%

The confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Blend 4 and 2 extends from 4.743 to 14.257. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence interval for the difference between the means of Blend 2 and 1 extends from -10.924 to -1.409. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence interval for the difference between the means of Blend 4 and 3 extends from 0.326 to 9.841. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence intervals for all the remaining pairs of means include zero, which indicates that the differences are not statistically significant.
  • The 95% individual confidence level indicates that you can be 95% confident that each confidence interval contains the true difference for that specific comparison. However, the simultaneous confidence level indicates that you can be only 80.83% confident that all the intervals contain the true differences.

F-value

An F-value appears for each term in the Analysis of Variance table:
F-value for the model or the terms
The F-value is the test statistic used to determine whether the term is associated with the response.
F-value for the lack-of-fit test
The F-value is the test statistic used to determine whether the model is missing higher-order terms that include the predictors in the current model.

Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

A sufficiently large F-value indicates that the term or model is significant.

If you want to use the F-value to determine whether to reject the null hypothesis, compare the F-value to your critical value. You can calculate the critical value in Minitab or find the critical value from an F-distribution table in most statistics books. For more information on using Minitab to calculate the critical value, go to Using the inverse cumulative distribution function (ICDF) and click "Use the ICDF to calculate critical values".

Grouping

Use the grouping information table to quickly determine whether the mean difference between any pair of groups is statistically significant.

The grouping column of the Grouping Information table contains columns of letters that group the factor levels. Groups that do not share a letter have a mean difference that is statistically significant.

If the grouping table identifies differences that are statistically significant, use the confidence intervals of the differences to determine whether the differences are practically significant.

Interpretation

Grouping Information Using the Tukey Method and 95% Confidence
Paint
N
Mean
Grouping
A
A
B
A
B
B
Means that do not share a letter are significantly different.

In these results, the table shows that group A contains Blends 1, 3, and 4, and group B contains Blends 1, 2, and 3. Blends 1 and 3 are in both groups. Differences between means that share a letter are not statistically significant. Blends 2 and 4 do not share a letter, which indicates that Blend 4 has a significantly higher mean than Blend 2.

Histogram of residuals

The histogram of the residuals shows the distribution of the residuals for all observations.

Interpretation

Use the histogram of the residuals to determine whether the data are skewed or include outliers. The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
A long tail in one direction Skewness
A bar that is far away from the other bars An outlier

Because the appearance of a histogram depends on the number of intervals used to group the data, don't use a histogram to assess the normality of the residuals. Instead, use a normal probability plot.

A histogram is most effective when you have approximately 20 or more data points. If the sample is too small, then each bar on the histogram does not contain enough data points to reliably show skewness or outliers.

Individual value plot

An individual value plot displays the individual values in each sample. The individual value plot makes it easy to compare the samples. Each circle represents one observation. An individual value plot is especially useful when your sample size is small.

Interpretation

Use an individual value plot to examine the spread of the data and to identify any potential outliers. Individual value plots are best when the sample size is less than 50.

Skewed data

Examine the spread of your data to determine whether your data appear to be skewed. When data are skewed, the majority of the data are located on the high or low side of the graph. Skewed data indicate that the data might not be normally distributed. Often, skewness is easiest to detect with an individual value plot, a histogram, or a boxplot.

Right-skewed
Left-skewed

The individual value plot with right-skewed data shows wait times. Most of the wait times are relatively short, and only a few wait times are longer. The individual value plot with left-skewed data shows failure time data. A few items fail immediately, and many more items fail later.

Outliers

Outliers, which are data values that are far away from other data values, can strongly affect your results. Often, outliers are easy to identify on an individual value plot.

On an individual value plot, unusually low or high data values indicate potential outliers.

Try to identify the cause of any outliers. Correct any data-entry errors or measurement errors. Consider removing data values for abnormal, one-time events (special causes). Then, repeat the analysis.

Interval plot

Use the interval plot to display the mean and confidence interval for each group.

The interval plots show the following:
  • Each dot represents a sample mean.
  • Each interval is a 95% individual confidence interval for the mean of a group. You can be 95% confident that the group mean is within the group's confidence interval.
Important

Interpret these intervals carefully because your rate of type I error increases when you make multiple comparisons. That is, the more comparisons you make, the higher the probability that at least one comparison will incorrectly conclude that one of the observed differences is significantly different.

Interpretation

In these results, Blend 2 has the lowest mean and Blend 4 has the highest. You cannot determine from this graph whether any differences are statistically significant. To determine statistical significance, assess the confidence intervals for the differences of means.

Interval plot for differences of means

Use the confidence intervals to determine likely ranges for the differences and to assess the practical significance of the differences. The graph displays a set of confidence intervals for the difference between pairs of means. Confidence intervals that do not contain zero indicate a mean difference that is statistically significant.

Depending on the comparison method you chose, the plot compares different pairs of groups and displays one of the following types of confidence intervals.

  • Individual confidence level

    The percentage of times that a single confidence interval would include the true difference between one pair of group means if the study were repeated multiple times.

  • Simultaneous confidence level

    The percentage of times that a set of confidence intervals would include the true differences for all group comparisons if the study were repeated multiple times.

    Controlling the simultaneous confidence level is particularly important when you perform multiple comparisons. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

Interpretation

In these results, the confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Blend 4 and Blend 2 extends from 3.114 to 15.886. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence intervals for the remaining pairs of means all include zero, which indicates that the differences are not statistically significant.
  • The 95% simultaneous confidence level indicates that we can be 95% confident that all of these confidence intervals contain the true differences.
  • Each individual confidence interval has a confidence level of 98.89%. This result indicates that you can be 98.89% confident that each individual interval contains the true difference between a specific pair of group means. The individual confidence level for each comparison produce the 95% simultaneous confidence level for all six comparisons.
Tukey Simultaneous Tests for Differences of Means
Difference of Levels
Difference of Means
SE of Difference
T-Value
Adjusted P-Value
Individual confidence level = 98.89%

Mean

The mean of the observations within each group. The mean describes each group with a single value identifying the center of the data. It is the sum of all the observations with a group divided by the number of observations in that group.

Interpretation

The mean of each sample provides an estimate of each population mean. The differences between sample means are the estimates of the difference between the population means.

Because the difference between the group means are based on data from a sample and not the entire population, you cannot be certain it equals the population difference. To get a better sense of the population difference, you can use the confidence interval.

N

The sample size (N) is the total number of observations in each group.

Interpretation

The sample size affects the confidence interval and the power of the test.

Usually, a larger sample yields a narrower confidence interval. A larger sample size also gives the test more power to detect a difference. For more information, go to What is power?.

Normal probability plot of the residuals

The normal plot of the residuals displays the residuals versus their expected values when the distribution is normal.

Interpretation

Use the normal probability plot of residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.

Note

If your one-way ANOVA design meets the guidelines for sample size, the results are not substantially affected by departures from normality.

The following patterns violate the assumption that the residuals are normally distributed.

S-curve implies a distribution with long tails.

Inverted S-curve implies a distribution with short tails.

Downward curve implies a right-skewed distribution.

A few points lying away from the line implies a distribution with outliers.

If you see a nonnormal pattern, use the other residual plots to check for other problems with the model, such as missing terms or a time order effect. If the residuals do not follow a normal distribution and the data do not meet the sample size guidelines, the confidence intervals and p-values can be inaccurate.

Null hypothesis and Alternative hypothesis

One-way ANOVA is a hypothesis test that evaluates two mutually exclusive statements about two or more population means. These two statements are called the null hypothesis and the alternative hypotheses. A hypothesis test uses sample data to determine whether to reject the null hypothesis.

For one-way ANOVA, the hypotheses for the test are the following:
  • The null hypothesis (H0) is that the group means are all equal.
  • The alternative hypothesis (HA) is that not all group means are equal.

Interpretation

Compare the p-value to the significance level to determine whether to reject the null hypothesis.

Pooled StDev

The pooled standard deviation is an estimate of the common standard deviation for all levels. The pooled standard deviation is the standard deviation of all data points around their group mean (not around the overall mean). Larger groups have a proportionally greater influence on the overall estimate of the pooled standard deviation.

Interpretation

A higher standard deviation value indicates greater spread in the data. A higher value produces less precise (wider) confidence intervals and low statistical power.

Minitab uses the pooled standard deviation to create the confidence intervals for both the group means and the differences between group means.

Example of a pooled standard deviation

Suppose your study has four groups, as shown in the following table.
Group Mean Standard Deviation N
1 9.7 2.5 50
2 12.1 2.9 50
3 14.5 3.2 50
4 17.3 6.8 200

The first three groups are equal in size (n=50) with standard deviations around 3. The fourth group is much larger (n=200) and has a higher standard deviation (6.8). Because the pooled standard deviation uses a weighted average, its value (5.488) is closer to the standard deviation of the largest group.

P-value – Factor

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Interpretation

Use the p-value in the ANOVA output to determine whether the differences between some of the means are statistically significant.

To determine whether any of the differences between the means are statistically significant, compare the p-value to your significance level to assess the null hypothesis. The null hypothesis states that the population means are all equal. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
P-value ≤ α: The differences between some of the means are statistically significant
If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that not all of population means are equal. Use your specialized knowledge to determine whether the differences are practically significant. For more information, go to Statistical and practical significance.
P-value > α: The differences between the means are not statistically significant
If the p-value is greater than the significance level, you do not have enough evidence to reject the null hypothesis that the population means are all equal. Verify that your test has enough power to detect a difference that is practically significant. For more information, go to Increase the power of a hypothesis test.

Residuals versus fits

The residuals versus fits graph plots the residuals on the y-axis and the fitted values on the x-axis.

Interpretation

Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern What the pattern may indicate
Fanning or uneven spreading of residuals across fitted values Nonconstant variance
A point that is far away from zero An outlier
The following graphs show an outlier and a violation of the assumption that the residuals are constant.
Plot with outlier

One of the points is much larger than all of the other points. Therefore, the point is an outlier. If there are too many outliers, the model may not be acceptable. You should try to identify the cause of any outlier. Correct any data entry or measurement errors. Consider removing data values that are associated with abnormal, one-time events (special causes). Then, repeat the analysis.

Plot with nonconstant variance

The variance of the residuals increases with the fitted values. Notice that, as the value of the fits increases, the scatter among the residuals widens. This pattern indicates that the variances of the residuals are unequal (nonconstant).

If you identify any patterns or outliers in your residual versus fits plot, consider the following solutions:

Issue Possible solution
Nonconstant variance Consider using Stat > ANOVA > One-Way in Minitab Statistical Software. Under Options, uncheck Assume equal variances.
An outlier or influential point
  1. Verify that the observation is not a measurement error or data-entry error.
  2. Consider performing the analysis without this observation to determine how it impacts your results.

Residuals versus order

The residual versus order plot displays the residuals in the order that the data were collected.

Interpretation

Use the residuals versus order plot to verify the assumption that the residuals are independent from one another. Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent. Ideally, the residuals on the plot should fall randomly around the center line:
If you see a pattern, investigate the cause. The following types of patterns may indicate that the residuals are dependent.
Trend
Shift
Cycle

Residuals versus the variables

The residual versus variables plot displays the residuals versus another variable. The variable could already be included in your model. Or, the variable may not be in the model, but you suspect it affects the response.

Interpretation

If you see a non-random pattern in the residuals, it indicates that the variable affects the response in a systematic way. Consider including this variable in an analysis.

R-sq

R2 is the percentage of variation in the response that is explained by the model. It is calculated as 1 minus the ratio of the error sum of squares (which is the variation that is not explained by model) to the total sum of squares (which is the total variation in the model).

Interpretation

Use R2 to determine how well the model fits your data. The higher the R2 value, the better the model fits your data. R2 is always between 0% and 100%.

You can use a fitted line plot to graphically illustrate different R2 values. The first plot illustrates a simple regression model that explains 85.5% of the variation in the response. The second plot illustrates a model that explains 22.6% of the variation in the response. The more variation that is explained by the model, the closer the data points fall to the fitted regression line. Theoretically, if a model could explain 100% of the variation, the fitted values would always equal the observed values and all of the data points would fall on the fitted line.
Consider the following issues when interpreting the R2 value:
  • R2 always increases when you add additional predictors to a model. For example, the best five-predictor model will always have an R2 that is at least as high the best four-predictor model. Therefore, R2 is most useful when you compare models of the same size.

  • Small samples do not provide a precise estimate of the strength of the relationship between the response and predictors. If you need R2 to be more precise, you should use a larger sample (typically, 40 or more).

  • R2 is just one measure of how well the model fits the data. Even when a model has a high R2, you should check the residual plots to verify that the model meets the model assumptions.

R-sq (adj)

Adjusted R2 is the percentage of the variation in the response that is explained by the model, adjusted for the number of predictors in the model relative to the number of observations. Adjusted R2 is calculated as 1 minus the ratio of the mean square error (MSE) to the mean square total (MS Total).

Interpretation

Use adjusted R2 when you want to compare models that have different numbers of predictors. R2 always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R2 value incorporates the number of predictors in the model to help you choose the correct model.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled potato chips per container. You receive the following results as you add the predictors in a forward stepwise approach:
Step % Potato Cooling rate Cooking temp R2 Adjusted R2 P-value
1 X     52% 51% 0.000
2 X X   63% 62% 0.000
3 X X X 65% 62% 0.000

The first step yields a statistically significant regression model. The second step adds cooling rate to the model. Adjusted R2 increases, which indicates that cooling rate improves the model. The third step, which adds cooking temperature to the model, increases the R2 but not the adjusted R2. These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.

R-sq (pred)

Predicted R2 is calculated with a formula that is equivalent to systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. The value of predicted R2 ranges between 0% and 100%.

Interpretation

Use predicted R2 to determine how well your model predicts the response for new observations. Models that have larger predicted R2 values have better predictive ability.

A predicted R2 that is substantially less than R2 may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population, although they may appear important in the sample data. The model becomes tailored to the sample data and therefore, may not be useful for making predictions about the population.

Predicted R2 can also be more useful than adjusted R2 for comparing models because it is calculated with observations that are not included in the model calculation.

For example, an analyst at a financial consulting company develops a model to predict future market conditions. The model looks promising because it has an R2 of 87%. However, the predicted R2 is only to 52%, which indicates that the model may be over-fit.

S

S represents the standard deviation of how far the data values fall from the fitted values. S is measured in the units of the response.

Interpretation

Use S to assess how well the model describes the response. S is measured in the units of the response variable and represents the standard deviation of how far the data values fall from the fitted values. The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled potato chips per container. You reduce the model to the significant predictors, and S is calculated as 1.79. This result indicates that the standard deviation of the data points around the fitted values is 1.79. If you are comparing models, values that are lower than 1.79 indicate a better fit, and higher values indicate a worse fit.

SE of Difference

The standard error of the difference between means (SE of Difference) estimates the variability of the difference between sample means that you would obtain if you took repeated samples from the same populations.

Interpretation

Use the standard error of the difference between means to determine how precisely the differences between the sample means estimate the differences between the population means. A lower standard error value indicates a more precise estimate.

Minitab uses the standard error of the difference to calculate the confidence intervals of the differences between means, which is a range of values that is likely to include the population differences.

Significance level

The significance level (denoted by alpha or α) is the maximum acceptable level of risk for rejecting the null hypothesis when the null hypothesis is true (type I error).

Interpretation

Use the significance level to decide whether to reject or fail to reject the null hypothesis (H0). When the p-value is less than the significance level, the usual interpretation is that the results are statistically significant, and you reject H0.

For one-way ANOVA, you reject the null hypothesis when there is sufficient evidence to conclude that not all of the means are equal.

Simultaneous CI of the difference

Use the simultaneous confidence intervals to identify mean differences that are statistically significant, to determine likely ranges for the differences, and to assess the practical significance of the differences. The table displays a set of confidence intervals for the difference between pairs of means. Confidence intervals that do not contain zero indicate a mean difference that is statistically significant.

The simultaneous confidence level is the percentage of times that a set of confidence intervals includes the true differences for all group comparisons if the study were repeated multiple times.

Controlling the simultaneous confidence level is particularly important when you perform multiple comparisons. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

The confidence interval of the difference is composed of the following two parts:

Point estimate
The point estimate is the difference between a pair of means and is calculated from the sample data. The confidence interval is centered around this value.
Margin of error
The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the error margin is added to the point estimate. To calculate the lower limit of the confidence interval, the error margin is subtracted from the point estimate. The margin widens as the number of comparisons increases in order to maintain the simultaneous level of confidence that all intervals contain the true population differences.

Interpretation

Use the confidence intervals to assess the differences between group means.

Tukey Simultaneous Tests for Differences of Means
Difference of Levels
Difference of Means
SE of Difference
T-Value
Adjusted P-Value
Individual confidence level = 98.89%

The confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Blend 2 and Blend 4 extends from 3.114 to 15.886. This range does not include zero, which indicates that the difference is statistically significant.
  • The confidence intervals for the remaining pairs of means all include zero, which indicates that the differences are not statistically significant.
  • The 95% simultaneous confidence level indicates that we can be 95% confident that all of these confidence intervals contain the true differences.
  • The 98.89% individual confidence level indicates that we can be 98.89% confident that each confidence interval contains the true difference for that specific comparison.

Standard Deviation (StDev)

The standard deviation is the most common measure of dispersion, or how spread out the data are around the mean. The symbol σ (sigma) is often used to represent the standard deviation of a population. The symbol s is used to represent the standard deviation of a sample.

Interpretation

The standard deviation uses the same units as the variable. A higher standard deviation value indicates greater spread in the data. A good rule of thumb for a normal distribution is as follows:
  • Approximately 68% of the values fall within one standard deviation of the mean.
  • 95% of the values fall within two standard deviations.
  • 99.7% of the values fall within three standard deviations.

The sample standard deviation of a group is an estimate of the population standard deviation of that group. The standard deviations are used to calculate the confidence intervals and the p-values. Larger sample standard deviations result in less precise (wider) confidence intervals and lower statistical power.

Analysis of variance assumes that the population standard deviations for all levels are equal. If you cannot assume equal variances, use Welch's ANOVA, which is an option for One-Way ANOVA that is available in Minitab Statistical Software.

T-value

The t-value in one-way ANOVA is a test statistic that measures the ratio between the difference in means and the standard error of the difference.

Interpretation

You can use the t-value to determine whether to reject the null hypothesis, which states that the difference in means is 0. However, the p-value is used more often because it is easier to interpret. For more information on using the critical value, go to Using the t-value to determine whether to reject the null hypothesis.

Minitab uses the t-value to calculate the p-value.

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