A coin is tossed 4 times.

How many heads would you expect to get,on average.

the expected value is a way of estimating the mean without resorting to actual experiments.

You would expect to get a head about half of the time

we expect

1

---- * 4=2

2

Standard Deviation

The range is the difference between the highest and lowest scores in a data set and is the simplest measure of spread. So we calculate range as:

The standard deviation is a measure of the spread of scores within a set of data

The population standard deviation formula is

= population standard deviation

= sum of...

= population mean

= number of scores in sample

we write the expectation of X as E(X)

Range

Range = maximum value - minimum value

For example, let us consider the following data set:

23 56 45 65 59 55 62 54 85 25

What is the range?

The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculated differently. In statistics, we are usually presented with having to calculate sample standard deviations, and so this is what this article will focus on, although the formula for a population standard deviation will also be shown.

The maximum value is 85 and the minimum value is 23. This results in a range of 62, which is 85 minus 23

maximum value :85

minimum value: 23

85 - 23 = 62

So the range is 62

**measures of spread**

Basic examples

For a finite set of numbers, the standard deviation is found by taking the square root of the average of the squared differences of the values from their average value

consider a population consisting of the following eight values:

2 4 4 4 5 5 7 9

These eight data points have the mean (average) of 5

2＋4+4+4+5+5+7+9

－－－－－－－－－－－－－－－－－－－－－－－－－－－ ＝5

8

First, calculate the difference of each data point from the mean, and square the result of each

Next, calculate the mean of these values, and take the square root:

This quantity is the population standard deviation, and is equal to the square root of the variance. This formula is valid only if the eight values we began with form the complete population. If the values instead were a random sample drawn from some larger parent population, then we would have divided by 7 (which is n−1) instead of 8 (which is n) in the denominator of the last formula, and then the quantity thus obtained would be called the sample standard deviation. Dividing by n−1 gives a better estimate of the population standard deviation than dividing by n.

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