Stats question

Clauric

Just a question that I can't find an answer to:

Questions answered on a scale of 1 - 5. Mean is 4.1. Standard deviation is .92

How can the standard deviation exceed the scalar amount, and what does it signify?

Yakuza

The standard deviation is a measure of how much each measurement deviates (differs) from the mean.
It is an average, so it doesn't have to conform to the possible values in the scale, much like the mean is an average of the values being studied.
The reason it is squared and then the square root taken is that the average deviation from the average is by definition zero, so simply adding up these values won't help. Squaring, summing and then taking the square root gives an average of the magnitude (size) of the deviation, which is what SD measures.

Here is a sample set of data that has the same average of your set and a similar standard deviation.

[B]Reading		Value		Value - Average		(Value - Average)²[/B]
1		3			-1.1			1.21
2		3			-1.1			1.21
3		3			-1.1			1.21
4		3			-1.1			1.21
5		4			-0.1			0.01
6		5			0.9			0.81
7		5			0.9			0.81
8		5			0.9			0.81
9		5			0.9			0.81
10		5			0.9			0.81
[B]Average		4.1	Average (Variance)			0.89
			Standard Dev				0.943398113
			(Square root of Variance)
[/B]

Intuitively, you can see that in the set above, most values (3 and 5) are in or around 1 away from the average (4.1), and the actual value of 0.94 proves this.


Here's another set with the same average but with a smaller standard deviation (as most values are very close to the mean)

[B]Reading		Value		Value - Average		(Value - Average)²[/B]
1		4			-0.1			0.01
2		4			-0.1			0.01
3		4			-0.1			0.01
4		4			-0.1			0.01
5		4			-0.1			0.01
6		4			-0.1			0.01
7		4			-0.1			0.01
8		4			-0.1			0.01
9		4			-0.1			0.01
10		5			0.9			0.81
[B]Average		4.1	Average (Variance)			0.09
			Standard Dev				0.3
			(Square root of Variance)	
[/B]