Statistics - Standard Deviation?

Question

Statistics - Standard Deviation?

Answer 1

Suppose lots of students went into do an exam. The more the better. On a national scale, if all the results were taken and plotted onto a graph, some students will do poorly, some will do very well and the majority will lie in between the two extremes.

By human nature, the graph plotted will be a normal distribution graph, it will look like a bell shape design. The highest point of the graph will be the mean, and at this point when a line is drawn down vertically to cut the x-axis, it will divide the graph in two equal parts.

This kind of structure is what's known as a normal distribution curve and it is symmetrical about it's centre line which coincides with the mean (x bar) of the observations.



The standard deviation is a measure of how this data is spread out.

One standard deviation of both sides of the mean (x bar ±σ) will contain 68% of the data.

Two standard deviation of both sides of the mean (x bar ±2σ) will contain 95% of the data.

Three standard deviation of both sides of the mean (x bar ±3σ) will contain 99∙7% of the data.

These values apply for a normal distribution. So the standard distribution is a measure of the spread of data from the mean.

Answer 2

An Standard Deviation usually have got two terms, the one that appears above is quite simple to understand. Imagine five ages 8, 2, 4, 1 and 3. The sum is 18. Then you get the average doing this 18/5=3.6 You have to go beyond.

See that (8-3.6) + (2-3.6) + (4-3.6) + (3 - 3.6) is almost an Standard Deviation. You have calculated the average age and you want to know the distance from each age to the average. If the result is high, data isn´t around average.

Answer 3

Basically, imagine that you've plotted a bunch of points on a graph. You then fit a line of best fit to it. Now, this line of best fit represents the average/expected value for any given x-coordinate. The SD is calculated by finding the difference between each of these points and the mean/expected value - and then finding the root-mean-square of each of these points.



The reason you find the root-mean-square rather than just the mean, is that the square (and thus the root) is required to eliminate the fact that the difference between some points and the mean will be a negative number.





Ultimately, you can think of the SD as the average difference between each point in a data set, and the mean.

This is a useful quantity, because the smaller the SD, the higher the correlation (this should be evident); likewise, the larger the SD, the less correlated the data.