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M E A N , R A N G E , S T A N D A R D D E V I A T I O N A N D E R R O R B A R S

Biomath

Mean and Range

Mean = Average (∑data/#of data points)

Range: measure of the spread of the data (difference between the largest and smallest values

Problem: Cheryl took 7 math tests in one marking period. What is the range of her test scores? 89, 73, 84, 91, 87, 77, 94

Solution: Ordering the test scores from least to greatest, we get: 73, 77, 84, 87, 89, 91, 94

highest - lowest = 94 - 73 = 21 Answer: The range of these test scores is 21 points.

Standard Deviation

A measure of how data points are spread out around the mean.

In a normal distribution, about 68% of all values lie within ±1 standard deviations of the mean. About 95% of all values lie within ±2 standard deviations from the mean.

The average American man is 5'10". This means half of all American men are taller than 5'10", and half are shorter than 5'10".

Suppose you measured the height of a hundred men chosen at random off the street. You would most likely measure something much like the following graph

Normal Distribution

Calculating Standard Deviation

Calculating Standard Deviation

1. Work out the Mean (the simple average of the numbers)

2. Then for each number: subtract the Mean and square the result

3. Then work out the mean of those squared differences.

4. Take the square root of that and you are done!

Calculating SD Example

Example: Sam has 20 Rose Bushes.

The number of flowers on each bush is

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

Work out the Standard Deviation.

Step 1: Find the mean

In the formula above μ (the Greek letter "mu") is the mean of all our values ...

Example: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

The mean is (9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7

So: μ = 7

Step 2: for each number: subtract the Mean and square the result

This is the part of the formula that says:

So what is xi ? They are the individual x values 9, 2, 5, 4, 12, 7, etc...

In other words x1 = 9, x2 = 2, x3 = 5, etc.

Step 2 Continued

(9 - 7)2 = (2)2 = 4

(2 - 7)2 = (-5)2 = 25

(5 - 7)2 = (-2)2 = 4

(4 - 7)2 = (-3)2 = 9

(12 - 7)2 = (5)2 = 25

(7 - 7)2 = (0)2 = 0

(8 - 7)2 = (1)2 = 1

... etc ...

Step 3. Then work out the mean of those squared differences.

To work out the mean, add up all the values then divide by how many.

The notation for "add up all the values" is "Sigma": Σ

Step 3 cont.

We want to add up all the values from 1 to N, where N=20 in our case because there are 20 values:

Which means: Sum all values from (x1-7)2 to (xN-7)2

We already calculated (x1-7)2=4 etc. in the previous step, so just sum them up:

= 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9 = 178

Step 3 cont.

But that isn't the mean yet, we need to divide by how many, which is simply done by multiplying by "1/N":

Mean of squared differences = (1/20) × 178 = 8.9

(Note: this value is called the "Variance")

Step 4: Take the square root

σ = √(8.9) = 2.983...

Example 2

You and your friends have just measured the heights of your dogs (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Find out the Mean, the Variance, and the Standard Deviation.

Calculate the mean

Mean = (600 + 470 + 170 + 430 + 300)/5 = 1970/5

= 394

Now, we calculate each dogs difference from the Mean:

Now, we calculate each dogs difference from the Mean:

2. Then for each number: subtract the Mean and square the result

3. Then work out the mean of those squared differences.

Standard Deviation is just the square root of our last value

Standard Deviation: σ = √21,704 = 147.32... = 147(to the nearest mm)

And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation (147mm) of the Mean:

Error Bars

Error bars show either the range of the data or the standard deviation on a graph.

Error bars and Significant Difference

Two data sets are significantly different if the difference in the numbers is too large to be caused by random chance.

If error bars overlap, the difference between the data sets is likely NOT significant.

IF error bars do not overlap, then the difference between the data sets is likely significant

Practice Question 1

What is the standard deviation for the numbers: 75, 83, 96, 100, 121 and 125?(Try to do this yourself, without using the Standard Deviation function on your calculator.)

A 16.9

B 17.1

C 17.6

D 18.2

Practice Question 2

Ten friends scored the following grades in their end-of-year math exam:23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%

What was the standard deviation of their marks?

(Try to do this yourself, without using the Standard Deviation function on your calculator.)

A 15.1% B 15.5% C 16.9% D 18.6%

Practice Question 3

A booklet has 12 pages with the following numbers of words:271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314

What is the standard deviation number of words per page?

A 22.6

B 22.0

C 21.9

D 21.4