How to calculate standard deviation?

Question

x(i) | freq.

8 | 11

10 | 9

12 | 13

14 | 24

16 | 16

18 | 10

20 | 15

The formula for standard deviation is $\sigma = \sqrt{\frac{\Sigma|x-\bar{x}|^2}{n}}$. It would be easy with a graphing calculator, but I only have TI-30XA scientific calculator that can't do much. Can someone teach me a faster way to calculate the standard deviation by hand?

Answer 1

The standard deviation $S=\sqrt{V},$ where $$V=\dfrac{1}{N}\displaystyle\sum_{i=1}^N(X_i-\bar{X})^2=\left(\dfrac{1}{N}\displaystyle\sum_{i=1}^NX_i^2\right)-\bar{X}^2=\left(\frac{1}{\sum_{j=1}^kn_j}\displaystyle\sum_{j=1}^kn_jX_j^2\right)-\left(\frac{1}{\sum_{j=1}^kn_j}\displaystyle\sum_{j=1}^kn_jX_j\right)^2.$$ In your case,

$$V=\frac{11\times 8^2+9\times 10^2+\dots 15\times 20^2}{11+9+\dots+15}-\left(\frac{11\times 8+9\times 10+\dots 15\times 20}{11+9+\dots+15}\right)^2.$$

Answer 2

I would have been surprised if something marketed by TI as a "scientific calculator" did not have an ability to do this calculation relatively easily. It seems that indeed it can, as shown on page 10 of the manual. You can find the manual on-line at http://pdfstream.manualsonline.com/7/7694cf74-0334-4ebb-8f1e-f901f463cab4.pdf.

1. Press $\fbox{2nd}$ [CSR] function to clear any previous data.

2. Key in a data value (just the number, no other keystrokes).

3. Press $\fbox{2nd}$ [FRQ], then key in the frequency of the data value you just entered.

4. Press $\Sigma+$ key. This records the data value and frequency.

5. Repeat steps 2 through 4 for each other data value.

6. Press $\fbox{2nd}$ [$\sigma_{x\,n}$] to compute the standard deviation according to the formula you wrote.

In your example the first few keystrokes would be

$\fbox{2nd}$ [CSR] $8$ $\fbox{2nd}$ [FRQ] $1$ $1$ $\Sigma+$

This is a black box approach, since it tells you nothing about how the formula is actually calculated.

You already have another answer showing an actual mathematical approach. The point of this is that you take the mean of the squares of the data values, subtract the square of the mean of the (unsquared) data, and take the square root. If the $x(i)$ were values of a discrete random variable $X$ you would write

$$ SD = \sqrt{E[X^2] - (E[X])^2}. $$