Starter.

Calculate the standard deviation for the following data set.

Monthly average Temperatures.

Month Temperature in degrees C

1 5

2 5

3 6

4 10

5 15

6 16

7 20

8 28

9 25

10 20

11 12

12 8

• Stage 1- Tabulate the values (x) and their squares (x ² ). Add these values (∑x and ∑x ² ).

• Find the mean of all the values of x (x ) and square it (x ² ).

• Stage 3- Calculate the formula

= ∑x² - x ²

n

Method.

Answer.

• Sum of X squared = 3084

• Mean squared = 201.6

• 3084 divided by 12 =257

• 257- 201.6= 55.4

• Square root of 55.4 = 7.4

• Standard deviation =7.4.

Confidence Limits.

• In a research project you usually collect sample data: 100 people, 100 houses, 100 stream measurements.

• It is helpful to estimate how close the results you get from measuring your samples are to the result you would get if you measured the total population.

• These are known as the confidence limits.

Calculating Confidence limits.

30 or more samples• If we wished to now measure pebbles on a

beach we would do the following.

Stage 1- Take a random sample (n)- 100 pebbles.

Stage 2- Mean length of this sample (x)- 50 mm

Stage 3 calculate the standard deviation of this data (s)- 10

Stage 4 Calculate the standard error of the sample mean (SEx )

Sample Error of the Mean

• SEx= s = 10 = 1.

n 10

From this sample we can now make the following statements.

Statements..

1. There is a 68% probability that the mean length of all the pebbles lies within one standard error of our sample mean, i.e 49 to 51mm

2. There is a 95% probability that the real mean is within two standard errors of our sample mean, i.e 48 to 52mm

3. There is a 99.7% chance that the real mean is within three standard errors of our sample mean i.e 47 to 53

Percentage probability.

• The percentage probability figure is known as the confidence level.

• The range of values within which the real mean might lie are the confidence limits.

• So what we are saying is that if I measured every pebble on the beach there is a 95% chance that the average length we would find would be between 48 and 52 mm.

Over to you.

• Calculate the 95 %confidence limits for the following data sets.

• Show your working for each calculation.

• SEx= s = 10 = 1.

n 10

Random sample

Mean length Standard deviation

100 26 10

100 82 10

100 66 10

100 42 10

Answers

1. 95% confident the mean lies between 24-28

2. 95 % confident the mean lies between 80 and 84

3. 95 % confident the mean lies between 64 and 68

4. 95 % confident the mean lies between 40 and 44

Tests of significance.

• What we have just looked at was confidence limits- what we are going to start looking at now are tests of significance.

Why do we have to do this?

• When carrying out a project you will often collect two or more sets of sample data with the aim of comparing them e.g

1. Land values at the centre and outskirts of a town.2. Temperatures in and out of a wood.3. Pebble size at each end of a beach4. Crop yields on two different rock types.• Tests of significance are used to tell us whether the

differences between two or more data sets of sample data are truly significant or whether these differences occurred by chance.

Example.

• If we measured the temperatures 20 times on a north facing slope and 20 times on a south facing the result might be

• North 13.4 degrees c• South 13.7 degrees c• Can we say with confidence that the actual

(rather than the sampled) temperatures are higher on the south facing slope?

• Or could it be that differences between the figures are due to chance and that another sample would give a different result.

Tests of significance.

• These tests tell us the probability that differences between sample data are due to chance.

• If we find that there is a significant probability that the relationship could have occurred due to chance this can mean one of two things.

1. The relationship is not significant and there is little point in looking further into it.

2. Our sample is too small. If we took a larger sample, we might find that the result of the test of significance changes: the relationship comes more certain

Good Geography.

• It is not possible to tell which of these two conclusions is the correct one from the result of the test itself.

• This is a good example of the way that statistics are only a tool and can never replace good geographical thinking.– Next week- looking at the tests of

significance- Mann Whitney U test and Chi Squared.

Homework

• Find a definition for the following.

• You will need this for later lessons. – Correlation– Positive and Negative Correlation – Spearmans Rank Correlation Coefficient (with

method if possible).