What is a really easy explanation of standard deviation?

Answer 1

Standard deviation tells you how spread out your set of values is compared with the average (of your set of values).

For example, if you have the heights of all the players in a soccer football team, then you can work out the average height (the mean). But if you know the mean, that doesn't tell you much about the spread. If the average height is 180 cm, you don't know if ALL the players were 180 cm, or if they were all between 175 and 195 cm. You don't know if one of them was 210 cm, or if some were really short. If we know the SPREAD then we have some extra information.

The standard deviation is the average difference between a player's height and the average for the team. So if the team average height is 180 cm, and the standard deviation is small, say 4 cm, then you know that most players are between 176 and 184 cm. If the standard deviation is large (say 18 cm) then most players are between 162 and 198 cm, a much bigger range!! So the standard deviation really does tell you something about your data.

Basically, standard deviation is when you measure the differences between your players and the average height. Some will be shorter than average (with a negative difference) and some will be taller than average (with a positive difference). And some may have a zero difference (if they are the same height as the mean).

If you add up all these differences, the negative ones will cancel out the positives, and you won't get any useful information. So you SQUARE all the differences first before you add them up. When you square a negative number it becomes positive (-2 times -2 = +4). Then you get the average of all the squared differences (add them all up and divide the number of answers, that is, eleven). So for our eleven players, square the difference between each one's height and the average. Add them all together, and divide by 11. This answer is called the VARIANCE.

(If you were only measuring a sample of the team you would divide by 10 [one FEWER than the total number], but because you measured the whole population of the team, you divide by 11.)

Get the square root of the variance (remember you squared all the numbers, now you unsquare them), and the answer is the standard deviation. (Square root is the opposite of squared. Four squared = 16. The square root of 16 is 4.)

Here it is again:

1. Get the average (mean) of the heights of all your players.
2. Work out all the differences between their heights and the average. Shorter players will have a negative difference, taller players will have a positive.
3. Square each difference (Square means multiply it by itself, eg, -8 x -8 = +64). All the answers will be positive.
4. Add all the answers together and divide by 11. This number is called the Variance.
5. Get the square root of the Variance and THAT is the Standard Deviation.

A small standard deviation (3 or 4 cm) tells you that most of the team are about the same size. A large standard deviation (15 to 20 cm) tells you that you have a bigger spread, and might have some really tall, and some really short. Answer:
The question actually asked for "a really easy explanation". Now, although it is not an easy concept for any really easy explanation, I'm sure we can simplify a little the great mass that we have above for the average 'JoeBlow'.
Standard deviation is, as mentioned above, a measure of "the spread", or how far spread apart, from the average of all the figures you are considering, or of all the set of measurements you have made about something.
To possess meaning, we express this 'spread' using numbers. 1 standard deviation, for instance, ABOVE the average, or mean, of all the values in your sample is the point at which 34% of the values nearest, but above the mean lie. On the other hand, the 34% of numbers closest to the mean, but Below the mean is called the -1 standard deviation value. So 68% of all the values in your sample fall inside 1 standard deviation above the mean and 1 standard deviation below the mean. This region will, therefore, possess the middle 68% of all the values in your sample - which is most of them really.