Definition/Introduction

The standard deviation (SD) measures the extent of scattering in a set of values, typically compared to the mean value of the set.[1][2][3] The calculation of the SD depends on whether the dataset includes the entire target population or merely samples it. Ideally, studies obtain data from an entire population, which defines the population parameter. However, this is rarely possible in medical research, so a population sample is often used.[4]

The sample SD is determined through the following steps:

1. Calculate the deviation of each observation (each data point) from the mean.
2. Square each of these results to remove negative values.
3. Add these new values together.
4. Subtract 1 from the total number of observations, then divide this number by the sum obtained above; this result is the sample variance.
5. Find the square root of the sample variance; this number will be the sample SD.[1]

The process of determining the sample SD of a data set is outlined in the example below, which will use the following data: 4, 5, 5, 5, 7, 8, 8, 8, 9, 10

First, calculate the mean of the data set by adding the value of each observation together and then dividing by the number of observations. The sum of our example values is 69, which is then divided by 10 to calculate a mean of 6.9. 

Follow the same steps as above to work out the standard deviation:

1. Subtract the mean from each observation to obtain the following values: -2.9, -1.9, -1.9, -1.9, 0.1, 1.1, 1.1, 1.1, 2.1, 3.1
2. Each value is squared, resulting in new values: 8.41, 3.61, 3.61, 3.61, 0.01, 1.21, 1.21, 1.21, 4.41, 9.61
3. The sum of these new values is 36.9.
4. The sample variance is calculated by dividing this sum by 10 minus 1: 36.9/9 = 4.1.
5. Finally, the square root of the sample variance is the sample standard deviation, which is 2.02 (to 2 decimal places).

For the population SD, the only difference is during step 4; the sample variance is divided by the total number of observations (without subtracting 1).

Using the same data set to calculate the SD (assuming the data set was the total population):

1. Subtract the mean from each observation to obtain the following values: -2.9, -1.9, -1.9, -1.9, 0.1, 1.1, 1.1, 1.1, 2.1, 3.1
2. Each value is squared to remove any negative values, resulting in new values: 8.41, 3.61, 3.61, 3.61, 0.01, 1.21, 1.21, 1.21, 4.41, 9.61
3. The sum of these new values is 36.9.
4. The population variance is calculated by dividing this sum by the total number of observations: 36.9/10 = 3.69
5. Finally, the square root of the sample variance is the population standard deviation, which is 1.92 (to 2 decimal places).

The greater the deviation of each value from the data set's mean, the greater the standard deviation; a smaller SD indicates the values are all relatively close to the mean.[2][5]

Issues of Concern

The method of calculating sample SD described above is highly accurate. However, this method becomes increasingly inefficient as the data set values increase, as the difference from the mean of every value must be calculated. This step can be bypassed using the following steps:

1. List the observed values of the data set in a column, or 'x.'
2. Find the 'sum of x' by adding all the values in this column together.
3. Square the value of the 'sum of x.'
4. Divide this result by the total number of observations to find a new value, or 'y'.
5. Square each value in the 'x' column.
6. Find the sum of these squared values.
7. Subtract 'y' from this sum.
8. If calculating the sample SD, divide this value by the total number of observations minus 1. If calculating the population SD, divide the value by the total number of observations. 
9. The square root of this result is the sample (or population) SD.

This method can quickly calculate the sample SD of a large data set, particularly if paired with a memory function-enabled calculator or an electronic data analysis program.

One mistake researchers sometimes make involves choosing the SD or the standard error of the mean (SEM) to be reported alongside the mean. The distinction between the SD and SEM is crucial but often overlooked. Authors may report the incorrect variable in their data. While the SD refers to the scatter of values around the sample mean, the SEM refers to the accuracy of the sample mean itself. The SEM is a measurement of the precision of the sample mean compared to the total population mean. In contrast to the SD, the SEM does not provide information on the scatter of the sample.[5]

Despite this difference, the SEM and SD are often incorrectly used as interchangeable variables. There have been many hypotheses for this, including a lack of understanding of the meaning of these statistical concepts, leading authors to report what they have seen other authors report. Consequently, multiple articles may publish the SEM in an improper context. Another reason is that the SEM is smaller than the SD; if presented alongside the mean, a smaller value erroneously suggests a higher precision.

This error can also be observed in the figures generated from the same analysis (eg, using the SEM will shorten the error bars). This can even confuse experienced readers who typically expect the SD to be paired with the mean.[5][6] In response to these issues, some journals only allow authors to present SD to remove any chance of an author mistakenly using the SEM in an inappropriate context.[6][7]

Clinical Significance

Using SD allows for a quick overview of a population that is distributed along a normal (Gaussian) pattern. In these populations, 1 SD covers 68% of observations, 2 SD covers 95%, and 3 SD covers 99.7%.[5][6][8][9] This adds greater context to the mean value reported in many studies.

A hypothetical example may help clarify this point. For example, a study looking at the effect of a new chemotherapy agent on life expectancy concludes that the agent increases life expectancy by 5 years.

Widely different data sets could be analyzed to obtain a mean of 5 years. For example, everyone in the sample could have shown an increase in life expectancy between 4 to 6 years. However, another data set that would satisfy this mean would be if half the sample showed no statistical increase in life expectancy while the other half experienced an increase of 10 years. Including an SD can help readers quickly resolve this ambiguity, as the former case would have a small SD while the latter would have a large SD. This allows readers to interpret the results of the study more accurately.

Furthermore, finding a “normal” range of measurements within medicine is often critical information. One example could be a range of laboratory values (such as complete blood count) expected if the measurement was conducted in healthy individuals. Often, this is the 95% reference range, where 2.5% of values will be below the reference range, and 2.5% of values will be greater than the reference range. This makes the SD extremely useful in data sets that follow a normal distribution because it can quickly calculate the range in which 95% of values lie.[10]

Another advantage of reporting the SD is that it reports the scatter within the data in the same units as the data itself. This is in contrast to the variance of the data set, which is equivalent to the square of the SD; hence, its units are the units of the data squared. Therefore, although the variance can be useful in certain scenarios, it is generally not used when describing data. By sharing the same unit, the SD allows the data to be more easily interpreted.[1][2]

Clinicians should also know the disadvantages of using standard deviation. The main issue is in data sets where there are extreme values or severe skewness, as these results can influence the mean and SD by a significant amount. Consequently, in scenarios where the data set does not follow a normal (Gaussian) distribution, other measures of dispersion are often used. Most commonly, the interquartile range (IQR) is used alongside the median of the dataset. This is due to the IQR being significantly more resistant to extreme values, as only the data between the first and third quartiles are factored in when calculating the IQR. The data between the first and third quartile represents the middle 50% of values, so any unusually high or low values will not affect the calculation of the IQR. This gives a more accurate picture of the data set’s distribution than the SD.[1][11][12][13]

Nursing, Allied Health, and Interprofessional Team Interventions

Evidence-based medicine plays a significant role in patient care and relies on integrating individual clinical expertise with the best available literature.[14] Keeping up to date with the current literature is vital for all members of an interprofessional team to ensure they are acting in the best interest of a patient. To do this, all members must understand descriptive statistics (eg, mean, median, mode, standard deviation). Without this knowledge, it can be challenging to interpret the conclusions of research articles accurately.