What Is Standard Deviation? Variance, Calculation, the Normal Distribution

Test scores, product prices, daily temperatures: our world is full of collections of numbers (data). When we talk about such data, we often reach for the "mean" (average). But the mean alone does not tell us whether the data cluster near the average or are spread far apart. The single number that captures this "spread" is the standard deviation. This article explains, step by step, what standard deviation is, how it relates to variance, and how to calculate it.

What Is Standard Deviation

Standard deviation is a measure of how far data are spread from the mean. It is denoted by the lowercase Greek letter σ (sigma) or by s. The larger the value, the more widely the data spread from the mean; the smaller it is, the more tightly the data cluster near the mean.

Suppose two classes both averaged 60 on a test. If one class scored between 55 and 65 while the other ranged from 20 to 100, the same average hides very different realities. Standard deviation lets you express that difference numerically. While the mean describes the "center" of the data, the standard deviation describes its "spread" — think of them as a matched pair.

To see how the mean and standard deviation come out for real data, the quickest approach is to paste your numbers into the Descriptive Statistics Calculator. It shows the mean, variance, standard deviation, and more together, so you can understand the relationships at a glance.

The Relationship Between Variance and Standard Deviation

To understand standard deviation, you cannot skip variance. In fact, standard deviation is computed from variance. Their relationship is very simple: standard deviation is the square root of variance. Conversely, squaring the standard deviation gives the variance.

standard deviation = √variance　／　variance = standard deviation²

So what is variance? It is the average of the squared distances of each data point from the mean. The difference from the mean (called the deviation) is positive when above the mean and negative when below. Adding these up directly would cancel out to zero, so we square the differences to remove the sign, then take the average. That is the variance.

However, because variance squares the differences, its unit becomes the square of the original data (for scores, "points²"), which makes its size hard to grasp intuitively. Taking the square root returns the unit to the original, and that is the standard deviation. Because standard deviation lets you describe spread in the same unit as the original data (points, dollars, cm), it is used more often than variance in practice.

How to Calculate Standard Deviation

Standard deviation can be calculated with the following steps. Here we use the five values "2, 4, 4, 4, 6" as an example.

• Step 1: Find the mean. Add everything and divide by the count. (2+4+4+4+6)÷5 = 20÷5 = 4. The mean is 4.
• Step 2: Find each deviation (difference from the mean). They are -2, 0, 0, 0, 2.
• Step 3: Square each deviation. They become 4, 0, 0, 0, 4.
• Step 4: Average the squares (= variance). (4+0+0+0+4)÷5 = 8÷5 = 1.6. The variance is 1.6.
• Step 5: Take the square root of the variance. √1.6 ≈ 1.26. That is the standard deviation.

One caution: in Step 4 we said "divide by the count," which is the calculation for the population standard deviation, treating the data in hand as the whole. When you want to estimate the spread of a larger whole (the population) from a limited sample, it is common to divide not by n but by n-1 (using the unbiased variance). The fact that the divisor changes with your purpose, even for the same data, is a common source of confusion, so it is worth remembering.

Hand calculation is useful for understanding the steps, but it becomes tedious as data grow. For real work, just enter your numbers in bulk into the Descriptive Statistics Calculator to compute the mean, variance, and standard deviation at once. It shows both the population and sample (unbiased) standard deviation, so you can pick the value that fits your purpose.

What Large vs Small Standard Deviation Means

The value of the standard deviation by itself does not indicate "good" or "bad." It simply shows the size of the spread, and it must be interpreted together with the nature and purpose of the data.

In quality control, for example, a smaller standard deviation of dimensions means "little variation between units, so the process is stable," which is desirable. In investing, on the other hand, the standard deviation of returns is used as a measure of "risk (the size of fluctuation)," with larger values interpreted as more volatile. Thus the same "large standard deviation" can carry different meanings depending on context.

When you want to compare spread across data with different means, dividing the standard deviation by the mean gives the coefficient of variation (CV), which allows a fair comparison. It is a handy idea when lining up data of different units or scales.

The Normal Distribution and the 68-95-99.7 Rule

Standard deviation is especially powerful when data follow a normal distribution (a symmetric bell-shaped distribution). Many data in nature and society, such as test scores and heights, are known to roughly approximate a normal distribution.

In a normal distribution, once the mean and standard deviation are fixed, the proportion of data falling in any range is determined. This is captured by the famous 68-95-99.7 rule (an empirical rule).

• About 68% of the data fall within mean ± 1σ
• About 95% fall within mean ± 2σ
• About 99.7% fall within mean ± 3σ

For example, if a test with a mean of 60 and a standard deviation of 10 is close to normal, you can estimate that about 68% of test-takers score between 50 and 70, and about 95% between 40 and 80. Conversely, a value beyond 3σ from the mean (30 or 90 in this example) can be judged a "rare result" that seldom appears. Standard deviation is used to set defect thresholds in quality control, and "mean ± 3σ" is used as a rough guide for detecting outliers, precisely because of this property.

The Link to Standard Scores

The standard score (in Japan, the familiar "hensachi") is also built from the standard deviation. It is a value that converts where a data point sits within a group so that the mean becomes 50 and the standard deviation becomes 10. The formula is as follows.

standard score = (individual score − mean) ÷ standard deviation × 10 + 50

At the heart of this formula is the operation of dividing the difference from the mean by the standard deviation. This makes it possible to compare positions on the same yardstick even when tests differ in difficulty (in their mean or spread). A score exactly at the mean gives a standard score of 50; one σ above gives 60; two σ above gives 70.

Combined with the 68-95-99.7 rule above, you can get a feel for these scores. If the data are close to normal, a standard score of 60 or higher (mean + 1σ or more) corresponds to roughly the top 16%, and 70 or higher (mean + 2σ or more) to roughly the top 2.3%. Once you see that the standard score is a "relative position indicator" based on the standard deviation, its meaning becomes far more intuitive.

Standard deviation is a foundational statistical measure that captures, in a single number, the "spread" that the mean alone cannot reveal. Start by entering your own data in the Descriptive Statistics Calculator and see how the mean, variance, and standard deviation change. Watching the numbers move is the shortcut to understanding.

Please note that this article is intended to provide general information. The interpretation of statistical measures and the appropriate calculation method vary with the nature of the data and the purpose of the analysis. When making important decisions, decide after checking the latest official information, specialized references, and professional advice where needed.