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In the 1940s, Stanley Smith Stevens introduced four scales of measurement: nominal, ordinal, interval, and ratio. These are still widely used today as a way to describe the characteristics of a variable. Knowing the scale of measurement for a variable is an important aspect in choosing the right statistical analysis.

The “e” notation is scientific notation. For example, 1.23e-5 means 1.23 × 10⁻⁵ = 0.0000123. Prism uses this when numbers are very small (like P values) or very large to keep output readable.

A negative exponent (e.g., e-5) means the decimal point moves to the left, making the number smaller. A positive exponent moves it to the right. This notation is standard across scientific software and should be interpreted the same way as in any calculator or spreadsheet.

How to calculate the standard deviation

1. Compute the square of the difference between each value and the sample mean.
2. Add those values up.
3. Divide the sum by n-1. This is called the variance.
4. Take the square root to obtain the Standard Deviation.

Why n-1?

Why divide by n-1 rather than n? In step 1, you compute the difference between each value and the mean of those values. You don't know the true mean of the population; all you know is the mean of your sample. The data will be closer to the sample mean than to the true population mean, so the value you compute in step 2 will probably be a bit smaller than it would be if you used the true population mean. To compensate, divide by n-1 rather than n. This is called Bessel's correction. Statisticians say there are n-1 degrees of freedom.

When should SD be computed with n?

Use n-1 when you are analyzing a sample and wish to make more general conclusions, which is your best estimate for the population SD. Use n only if you simply want to quantify variation in a particular set of data with no intention to extrapolate. The goal of science is always to generalize, so n-1 is almost always the correct choice. GraphPad Prism always uses n-1.

It is tempting to look at whether two error bars overlap or not and try to reach a conclusion about whether the difference between means is statistically significant. Resist that temptation.

SD error bars quantify the scatter among the values. Looking at whether SD error bars overlap or not does not let you conclude whether the difference between means is statistically significant or not.

SEM error bars quantify how precisely you know the mean. If two SEM error bars do overlap and the sample sizes are equal or nearly equal, the P value is (much) greater than 0.05. The opposite rule does not apply: non-overlapping SEM bars could have P either above or below 0.05.

95% CI error bars are wider than SEM bars. If two 95% CI bars do not overlap and sample sizes are nearly equal, the difference is statistically significant with P much less than 0.05.

Rules of thumb apply only when sample sizes are equal or nearly equal.

Don't overemphasize P values

• Consider emphasizing the effect size and confidence interval, rather than a P value.
• Don't just say if the P value is greater or less than 0.05. If you can, give the P value as a number.
• It doesn't help to report a P value unless you clearly state what test was used.
• If you computed many P values, show them all and state the number of comparisons made.

Asterisks

Reporting style comparison

GraphPad Prism always reports a zero before the decimal point and four digits after. If the P value is less than 0.0001, it reports “<0.0001”.

If you want to show the variation in your data:

With fewer than 100 or so values, create a scatter plot that shows every value. This is the best way to show variation. If your data set has more than 100 values, use a box-and-whiskers plot, frequency distribution histogram, or cumulative frequency distribution. A graph showing mean and SD error bar is less informative than any of these alternatives.

If you want to show how precisely you have determined the mean:

Plot the 95% confidence interval of the mean. SEM error bars are shorter and appear more precise, but are harder to interpret than a confidence interval. Whatever error bars you show, always explicitly state your choice in the figure legend.

A note on misuse:

If your goal is to emphasize small differences, show SEM and hope readers think they are SD. If your goal is to cover up large differences, show SD and hope readers think they are SEM. Of course, this is dishonest, so always label your error bars correctly.

Excel uses the term “100% stacked column” graph to refer to a stacked bar graph where the sum of all slices is 100. Prism 6+ lets you make this in two ways:

1. As a parts of whole graph. Plots data entered into one column. Prism doesn't show any axis, giving a visual sense of division. If you want several stacks, make them individually and combine on a layout.
2. As a stacked bar graph (generally better). Enter data onto a Grouped table. Each row becomes one stack. Use the “Fraction of total” analysis (Prism 6+), choose to divide each value by its row total, and report results as percentages. Then plot the results.

In Prism 5, use the second approach only. There is no Fraction of Total analysis, so use Normalize, Transform, Row means, or Remove baseline analyses instead.

Transposing means every row becomes a column, and every column becomes a row. There are three ways to transpose in Prism:

1. From the data table, click Analyze, then choose Transpose from the list of data manipulations. The transposed data will appear on a new results table.
2. Copy a block of data to the clipboard. Put the insertion point at the upper-left corner of the target block. Choose Paste Transpose from the Edit menu or right-click shortcut menu.
3. When importing data, choose to Transpose in the Placement tab of the Import dialog.

To swap X and Y columns (not a full transpose), use the Transform analysis.

An odds ratio plot (Forest plot / meta-analysis plot) graphs odds ratios with 95% confidence intervals from several studies. GraphPad Prism can make this kind of graph easily.

1. Create a Column data table with replicates entered into columns.
2. Enter in each column: the odds ratio itself, plus the high and low confidence limits. Each treatment group gets a separate column with three values (the order of those three values doesn't matter). Label groups using column titles.
3. Click the Graphs tab and choose the thumbnail for your desired look. Choose to plot the median and range. The median of the three values is the point estimate, the range covers the confidence interval.
4. The default graph is vertical. To make it horizontal (as is standard for Forest plots), double-click the graph to open Format Graph, then go to the third tab.
5. To make the x-axis logarithmic (common in epidemiology, making odds ratios >1 and <1 symmetrical), format the x-axis with a Log 10 scale. Add a custom dotted grid line at X=1.0 via Format Axes.

Keywords: horizontal error bars, odds ratio, meta-analysis, Forest plot

Starting with Prism 8, you can choose which decimal format Prism uses to report P values (APA, NEJM, or GraphPad style). The standard symbol meanings are:

APA and NEJM styles show at most three asterisks (***). GraphPad style shows four asterisks (****) for P ≤ 0.0001. In the multiple t test analysis, current versions of Prism display “Yes” or “No” rather than asterisks.

You can interpret two-way ANOVA results by looking at the P values and multiple comparisons. The ANOVA table provides the underlying detail.

Sum-of-squares (SS)

• Interaction SS: Variation due to the fact that differences between rows are not the same for all columns (and vice versa).
• Row SS: Variation due to systematic differences between rows.
• Column SS: Variation due to systematic differences between columns.
• Residual SS: Variation not explained by any other source (error). Smaller when repeated measures are assumed.
• Total SS: Total variation among all values. Always the same regardless of repeated measures assumptions.

Degrees of freedom (df)

• Total df = (number of values) – 1
• Interaction df = (columns – 1) × (rows – 1)
• Row df = rows – 1; Column df = columns – 1

Mean squares (MS)

Each MS = SS / df for that row.

F ratio

Each F = MS(effect) / MS(denominator). The denominator depends on the design: for no repeated measures, it is always MSresidual. For repeated measures designs, different denominators apply for different factors (see Maxwell and Delaney for details).

P values

Each F ratio has a numerator df and denominator df. The P value is calculated from the F distribution with those two df values. In Excel: =FDIST(F, dfn, dfd).

The standard unpaired t test assumes that the two populations have identical standard deviations. As part of the t test analysis, Prism tests this assumption using an F test to compare variances.

What to do if variances differ

1. Conclude that the populations are different. Different standard deviations means the populations are different regardless of means. This may itself be the most important conclusion.
2. Transform your data. Logs are especially useful for lognormal data. Square root or reciprocal transforms may also equalize SDs.
3. Ignore the result (with caution). With roughly equal, moderately large sample sizes, the t test is robust to unequal variances.
4. Use Welch's t test. This corrects for unequal variances. In Prism, select “Welch's correction” under unpaired t test options. Note: it is not valid to use an F test first to decide whether to use Welch; this inflates Type I error. Welch must be pre-specified as part of your design.
5. Use a permutation test. No GraphPad program offers this, but it produces valid P values when populations have different standard deviations.

How to avoid the problem

Think clearly about the distribution of your data beforehand. If you know data are lognormal, always analyze logarithms. Alternatively, use Welch's t test routinely: you lose some power when SDs are equal but gain power when they are not (Ruxton recommendation).

After fitting a dose-response curve in Prism, go to the results sheet and use Analyze > Interpolate from curve. Enter the Y value corresponding to 90% response to obtain the EC90 directly from your fitted model.

You can use this approach for any EC value: EC10, EC20, EC75, etc. Just enter the appropriate Y value for your desired response level. Prism interpolates from the fitted curve, so the result accounts for all the parameters of your model.