Statistics Question: Standard Deviation?

Mean = 51.336

Variance = sum[(x-51.336)^2] / 149
= [ 24*(44.3-51.336)^2 + 27*(47.9-51.336)^2 + 42*(51.5-51.336)^2 + 45*(55.1-51.336)^2 + 12*(58.45-51.336)^2] / 149
= 4.298

Here’s another equation for Variance:

Variance= E(X^2)-E(X)^2, where E(X) is just the expected value or “mean.”

We get
E(X^2)= 24/150*44.3^2+27/150*47.9^2+…
+12/150*58.45^2=2653.73

You already calculated the mean,
So, E(X)=51.3
Variance=2653.73-(51.3)^2
=22.05

Then you square root the variance to get std. dev.
std. dev.= 4.696

Note the slightly different answer above. The answer isn’t incorrect he just used a different formula. He gave the sample std. dev. while I gave the true std. dev. Which formula you can use depends on if you’re collecting data from a sample. Also, I rounded the mean to the answer that you gave.

Variance is just the standard deviation squared. In other words, the square root of the varience is the standard deviation. The formula for the standard deviation is:

standard deviation = square root[ sigma(each x - mean)^2 / n ]

I promise it’s way easier than the formula suggests.

The sigma(each x – mean) ^2 just means that you take a value of x, subtract the mean from it, and square the difference. Do this for each value of x, individually. Then add up all of the squared differences.

Plugging in the values, with proper frequency you have: square root [ ( (44.3 - 51.3)^2 x 24 + (47.9 - 51.3)^2 x 27 + (51.5 - 51.3)^2 x 42 + (55.1 - 51.3)^2 x 45 + (58.45 - 51.3)^2 x 12 ) / 150 ]

The ” x 24, 27, etc.” is to take frequency into account, and makes it quicker to calculate that having to write out each value.

Some people find it easier to break up the formula. Take the sum of the squared differences, then divide by n, then take the square root.

In any case, you should get 4.28 as the standard deviation.

Feel free to ask any more questions if you do not understand. I know the formula can be hard to understand when written out.