SD and CV

The standard deviation (SD) quantifies variability. If the data follow a bell-shaped Gaussian distribution, then 68% of the values lie within one SD of the mean (on either side) and 95% of the values lie within two SD of the mean. The SD is expressed in the same units as your data.

Prism calculates the SD using the equation below. (Each y_i is a value, y_mean is the average, and N is sample size).

The standard deviation computed this way (with a denominator of N-1) is called the sample SD, in contrast to the population SD which would have a denominator of N.  Why is the denominator N-1 rather than N? In the numerator, 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.  Except for the rare cases where the sample mean happens to equal the population mean, the data will be closer to the sample mean than it will be to the population mean. This means that the numerator will be too small. So the denominator is reduced as well. It is reduced to N-1 because that is the number of degrees of freedom in your data. There are N-1 degrees of freedom, because you could calculate the remaining value from N-1 of the values and the sample mean.  

The coefficient of variation (CV) equals the standard deviation divided by the mean (expressed as a percent). Because it is a unitless ratio, you can compare the CV of variables expressed in different units. It only makes sense to report CV for variables where zero really means zero, such as mass or enzyme activity. Don't calculate CV for variables, such as temperature, where the definition of zero is somewhat arbitrary.

SEM

The standard error of the mean (SEM) is a measure of how far your sample mean is likely to be from the true population mean. The SEM is calculated by this equation:

With large samples, the SEM is always small. By itself, the SEM is difficult to interpret. It is easier to interpret the 95% confidence interval, which is calculated from the SEM.

95% Confidence interval

The confidence interval quantifies the precision of the mean. The mean you calculate from your sample of data points depends on which values you happened to sample. Therefore, the mean you calculate is unlikely to equal the overall population mean exactly. The size of the likely discrepancy depends on the variability of the values (expressed as the SD) and the sample size. Combine those together to calculate a 95% confidence interval (95% CI), which is a range of values. You can be 95% sure that this interval contains the true population mean. More precisely, if you generate many 95% CIs from many data sets, you expect the CI to include the true population mean in 95% of the cases and not to include the true mean value in the other 5% of the cases. Since you don't know the population mean, you'll never know when this happens.

The confidence interval extends in each direction by a distance calculated from the standard error of the mean multiplied by a critical value from the t distribution. This value depends on the degree of confidence you want (traditionally 95%, but it is possible to calculate intervals for any degree of confidence) and on the number of degrees of freedom in this experiment (N-1). With large samples, this multiplier equals 1.96. With smaller samples, the multiplier is larger.

Quartiles and range

Quartiles divide the data into four groups, each containing an equal number of values. Quartiles divided by the 25th percentile, 50th percentile, and 75th percentile. One quarter of the values are less than or equal to the 25th percentile. Three quarters of the values are less than or equal to the 75th percentile. The median is the 50th percentile.

Prism computes percentile values by first computing R= P*(N+1)/100, where P is 25, 50 or 75 and N is the number of values in the data set. The result is the rank that corresponds to the percentile value. If there are 68 values, the 25th percentile corresponds to a rank equal to .25*69=17.25. So the 25th percentile lies between the value of the 17th and 18th value (when ranked from low to high). Prism computes the 25th percentile as the average of those two values (some programs report the value one quarter of the way between the two).

Geometric mean

The geometric mean is the antilog of the mean of the logarithm of the values. It is less affected by outliers than the mean.

Which error bar should you choose?

It is easy to be confused about the difference between the standard deviation (SD) and standard error of the mean (SEM).

The SD quantifies scatter - how much the values vary from one another.

The SEM quantifies how accurately you know the true mean of the population. The SEM gets smaller as your samples get larger. This makes sense, because the mean of a large sample is likely to be closer to the true population mean than is the mean of a small sample.

The SD does not change predictably as you acquire more data. The SD quantifies the scatter of the data, and increasing the size of the sample does not increase the scatter. The SD might go up or it might go down. You can't predict. On average, the SD will stay the same as sample size gets larger.

If the scatter is caused by biological variability, your probably will want to show the variation. In this case, graph the SD rather than the SEM. You could also instruct Prism to graph the range, with error bars extending from the smallest to largest value. Also consider graphing every value, rather than using error bars.

If you are using an in vitro system with no biological variability, the scatter can only result from experimental imprecision. In this case, you may not want to show the scatter, but instead show how well you have assessed the mean. Graph the mean and SEM or the mean with 95% confidence intervals.

Ideally, the choice of which error bar to show depends on the source of the variability and the point of the experiment. In fact, many scientists always show the mean and SEM, to make the error bars as small as possible.