Standard Deviation Formula and Variance, With Solved Tables

The standard deviation formula, symbol by symbol

Finding the standard deviation using average formulas in maths is a two-step job: first average the values, then average the squared distances from that average. So the variance formula in maths is the core, and the standard deviation math formula is just its square root. Read the formula from the inside out. Each piece has one job.

• $x_i$ is one observation, the i-th value in your list.
• $\bar{x}$ (read "x bar") is the arithmetic mean of the data. See the mean, median and mode page if you need a refresher on finding it.
• $x_i-\bar{x}$ is the deviation of that value from the mean. Some deviations are negative, and the deviations always add up to zero, which is why we square them.
• $\sum$ means add the squared deviations for all values.
• $n$ is the number of values. Dividing by $n$ gives the variance, the average squared deviation.
• The square root brings the answer back to the same unit as the data. If the data is in kg, the standard deviation is in kg, and the variance is in kg squared.

The Greek letter $\sigma$ (sigma) is used for the population standard deviation and $\sigma^2$ for its variance. Many books use $s$ for the sample version. A small standard deviation means the values are packed close to the mean. A large one means they are spread out.

Population vs sample: n or n minus 1

School textbooks in India, including NCERT class 11, divide by $n$. They treat the data you are given as the whole group. In college statistics, economics and most exam questions that say "sample", you divide by $n-1$.

Why $n-1$? A sample's own mean sits closer to its values than the true population mean does, so the squared deviations come out a little too small. Dividing by one less than $n$ corrects this. Statisticians call $n-1$ the degrees of freedom. For large $n$ the two answers are almost the same.

Worked example: standard deviation of ungrouped data

The shortcut formula for variance

Expanding the square in the definition gives a version that skips the deviation column:

In words: the mean of the squares minus the square of the mean. It is faster when the mean is a messy decimal such as 23.47, because you never subtract it from each value. The sample version of the shortcut is $s^2 = \dfrac{\sum x_i^2 - n\bar{x}^2}{n-1}$.

Grouped data: direct, assumed mean and step deviation methods

For a frequency table, each class is represented by its mid-value $x_i$ and every term is weighted by its frequency $f_i$. There are three routes, and all give the same answer:

• Direct method: $\sigma^2 = \dfrac{\sum f_i x_i^2}{N} - \left(\dfrac{\sum f_i x_i}{N}\right)^2$.
• Assumed mean method: pick a convenient middle value $A$, let $d_i = x_i - A$, then $\sigma^2 = \dfrac{\sum f_i d_i^2}{N} - \left(\dfrac{\sum f_i d_i}{N}\right)^2$. Shifting all values by $A$ does not change the spread.
• Step deviation method: also divide by the class width $h$: $u_i = \dfrac{x_i - A}{h}$, then $\sigma^2 = h^2\left[\dfrac{\sum f_i u_i^2}{N} - \left(\dfrac{\sum f_i u_i}{N}\right)^2\right]$.

The step deviation method formula is the same idea as the one used for the mean in class 10, where $\bar{x} = A + h\cdot\dfrac{\sum f_i u_i}{N}$. Here you also need the $f_i u_i^2$ column, and you multiply by $h^2$ at the end because variance is in squared units.

Mean deviation formula

Mean deviation uses absolute values instead of squares. It is the average distance of the values from a central value, usually the mean or the median:

For the data 4, 8, 6, 10, 12 with mean 8, the absolute deviations are 4, 0, 2, 2, 4. Their total is 12, so M.D. about the mean $= \dfrac{12}{5} = 2.4$. The median here is also 8, so the mean deviation about the median is the same. Notice that 2.4 is smaller than the SD of 2.83. Mean deviation is never larger than the population standard deviation.

Quartile deviation and coefficient of variation

Quartile deviation (also called the semi-interquartile range) looks only at the middle half of the data, so a few extreme values do not disturb it. For n sorted values, take $Q_1$ as the $\tfrac{n+1}{4}$-th value and $Q_3$ as the $\tfrac{3(n+1)}{4}$-th value.

Coefficient of variation compares spread across data sets with different means or units. A lower C.V. means more consistent data.

Covariance in brief

Variance measures how one variable spreads. Covariance measures how two variables move together. A positive value means that when x is above its mean, y also tends to be above its mean.

Example: x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5. Here $\bar{x} = 3$, $\bar{y} = 4$ and $\sum x_i y_i = 66$, so $\operatorname{cov} = \dfrac{66}{5} - 12 = 1.2$. Divide by $n-1$ for sample covariance, which gives 1.5. Dividing covariance by $\sigma_x \sigma_y$ gives the correlation coefficient, which you meet in statistics for economics.

Common mistakes and exam tips

• Forgetting the square root. Variance and SD are different answers; read which one the question wants.
• Using class limits instead of mid-values in grouped data. Always use $x_i = \tfrac{\text{lower}+\text{upper}}{2}$.
• Dividing by the number of classes instead of $N = \sum f_i$.
• In the step deviation method, forgetting to multiply the bracket by $h^2$ (or the root by $h$).
• Rounding the mean early. Keep fractions until the last step, or use the shortcut form.
• Using $n-1$ in a class 11 board answer where NCERT expects $n$, or the reverse in a college sample question.

A useful property for MCQs: adding a constant to every value leaves the SD unchanged, while multiplying every value by k multiplies the SD by $|k|$ and the variance by $k^2$. For other topics in the same revision session, the all maths formulas page lists the key results chapter by chapter, and the percentage formula page helps when the C.V. needs to be written as a percent.