Most tabulation programs, spreadsheets or other data management tools will calculate the SD for you. For practical purposes, the computation is not important. Instead, it is "standardized," a somewhat complex method of computing the value using the sum of the squares. However, it is not actually calculated as an average (if it were, we would call it the "average deviation").
*Technical disclaimer: thinking of the Standard Deviation as an "average deviation" is an excellent way of conceptionally understanding its meaning. It describes the distribution in relation to the mean. It is used purely as a descriptive statistic. SD generally does not indicate "right or wrong" or "better or worse" - a lower SD is not necessarily more desireable. A distribution with a low SD would display as a tall narrow shape, while a large SD would be indicated by a wider shape. Respondent:Īnother way of looking at Standard Deviation is by plotting the distribution as a histogram of responses. The Standard Deviation of 1.15 shows that the individual responses, on average*, were a little over 1 point away from the mean. In Rating "B", even though the group mean is the same (3.0) as the first distribution, the Standard Deviation is higher. The individual responses did not deviate at all from the mean. In the first example (Rating "A") the Standard Deviation is zero because ALL responses were exactly the mean value. Consider the following example showing response values for two different ratings. Two very different distributions of responses to a 5-point rating scale can yield the same mean. The distribution of responses is important to consider and the SD provides a valuable descriptive measure of this. Looking at the mean alone tells only part of the story, yet all too often, this is what researchers focus on. But the higher SD for reliability could indicate (as shown in the distribution below) that responses were very polarized, where most respondents had no reliability issues (rated the attribute a "5"), but a smaller, but important segment of respondents, had a reliability problem and rated the attribute "1". At first glance (looking at the means only) it would seem that reliability was rated higher than value. The mean for a group of ten respondents (labeled 'A' through 'J' below) for "good value for the money" was 3.2 with a SD of 0.4 and the mean for "product reliability" was 3.4 with a SD of 2.1.
Let's say you've asked respondents to rate your product on a series of attributes on a 5-point scale.
SD tells the researcher how spread out the responses are - are they concentrated around the mean, or scattered far & wide? Did all of your respondents rate your product in the middle of your scale, or did some love it and some hate it? Standard Deviation (often abbreviated as "Std Dev" or "SD") provides an indication of how far the individual responses to a question vary or "deviate" from the mean. They are often referred to as the "standard deviation of the mean" and the "standard error of the mean." However, they are not interchangeable and represent very different concepts. Both statistics are typically shown with the mean of a variable, and in a sense, they both speak about the mean. The following article is intended to explain their meaning and provide additional insight on how they are used in data analysis. Standard Deviation and Standard Error are perhaps the two least understood statistics commonly shown in data tables. Obtained above, directly from the combined sample.Standard Deviation and Standard Error are perhaps the two least understood statistics commonly shown in data tables. Gives $S_c = 34.02507,$ which is the result we
Numerical verification of correct method: The code below verifies that the this formula Updating archival information with a subsequent sample. In many statistical programs, especially when [In the code below we abbreviate this sum asĪlthough somewhat messy, this process of obtaining combined sample variances (and thus combined sample SDs) is used Continuing on from BruceET's explanation, note that if we are computing the unbiased estimator of the standard deviation of each sample, namely $$s = \sqrt X_i^2$ in a formulaĪnalogous to the last displayed equation.