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  1. 1. Starter. Calculate the standard deviation for the following data set.
  2. 2. Monthly average Temperatures. 8 12 12 11 20 10 25 9 28 8 20 7 16 6 15 5 10 4 6 3 5 2 5 1 Temperature in degrees C Month
  3. 3. <ul><li>Stage 1- Tabulate the values (x) and their squares (x ² ). Add these values (∑x and ∑x ² ). </li></ul><ul><li>Find the mean of all the values of x (x ) and square it (x ² ). </li></ul><ul><li>Stage 3- Calculate the formula </li></ul><ul><li> = ∑x² - x ² </li></ul><ul><li> n </li></ul>Method.
  4. 4. Answer. <ul><li>Sum of X squared = 3084 </li></ul><ul><li>Mean squared = 201.6 </li></ul><ul><li>3084 divided by 12 =257 </li></ul><ul><li>257- 201.6= 55.4 </li></ul><ul><li>Square root of 55.4 = 7.4 </li></ul><ul><li>Standard deviation =7.4. </li></ul>
  5. 5. Confidence Limits. <ul><li>In a research project you usually collect sample data: 100 people, 100 houses, 100 stream measurements. </li></ul><ul><li>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. </li></ul><ul><li>These are known as the confidence limits. </li></ul>
  6. 6. Calculating Confidence limits. <ul><li>30 or more samples </li></ul><ul><li>If we wished to now measure pebbles on a beach we would do the following. </li></ul><ul><li>Stage 1- Take a random sample (n)- 100 pebbles. </li></ul><ul><li>Stage 2- Mean length of this sample (x)- 50 mm </li></ul><ul><li>Stage 3 calculate the standard deviation of this data (s)- 10 </li></ul><ul><li>Stage 4 Calculate the standard error of the sample mean (SEx ) </li></ul>
  7. 7. Sample Error of the Mean <ul><li>SEx= s = 10 = 1. </li></ul><ul><li>n 10 </li></ul><ul><li>From this sample we can now make the following statements. </li></ul>
  8. 8. Statements.. <ul><li>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 </li></ul><ul><li>There is a 95% probability that the real mean is within two standard errors of our sample mean, i.e 48 to 52mm </li></ul><ul><li>There is a 99.7% chance that the real mean is within three standard errors of our sample mean i.e 47 to 53 </li></ul>
  9. 9. Percentage probability. <ul><li>The percentage probability figure is known as the confidence level. </li></ul><ul><li>The range of values within which the real mean might lie are the confidence limits. </li></ul><ul><li>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. </li></ul>
  10. 10. Over to you. <ul><li>Calculate the 95 %confidence limits for the following data sets. </li></ul><ul><li>Show your working for each calculation. </li></ul><ul><li>SEx= s = 10 = 1. </li></ul><ul><li>n 10 </li></ul>
  11. 11. Standard deviation Mean length Random sample 10 26 100 10 82 100 10 66 100 10 42 100
  12. 12. Answers <ul><li>95% confident the mean lies between 24-28 </li></ul><ul><li>95 % confident the mean lies between 80 and 84 </li></ul><ul><li>95 % confident the mean lies between 64 and 68 </li></ul><ul><li>95 % confident the mean lies between 40 and 44 </li></ul>
  13. 13. Tests of significance. <ul><li>What we have just looked at was confidence limits- what we are going to start looking at now are tests of significance. </li></ul>
  14. 14. Why do we have to do this? <ul><li>When carrying out a project you will often collect two or more sets of sample data with the aim of comparing them e.g </li></ul><ul><li>Land values at the centre and outskirts of a town. </li></ul><ul><li>Temperatures in and out of a wood. </li></ul><ul><li>Pebble size at each end of a beach </li></ul><ul><li>Crop yields on two different rock types. </li></ul><ul><li>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. </li></ul>
  15. 15. Example. <ul><li>If we measured the temperatures 20 times on a north facing slope and 20 times on a south facing the result might be </li></ul><ul><li>North 13.4 degrees c </li></ul><ul><li>South 13.7 degrees c </li></ul><ul><li>Can we say with confidence that the actual (rather than the sampled) temperatures are higher on the south facing slope? </li></ul><ul><li>Or could it be that differences between the figures are due to chance and that another sample would give a different result. </li></ul>
  16. 16. Tests of significance. <ul><li>These tests tell us the probability that differences between sample data are due to chance. </li></ul><ul><li>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. </li></ul>
  17. 17. <ul><li>The relationship is not significant and there is little point in looking further into it. </li></ul><ul><li>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 </li></ul>
  18. 18. Good Geography. <ul><li>It is not possible to tell which of these two conclusions is the correct one from the result of the test itself. </li></ul><ul><li>This is a good example of the way that statistics are only a tool and can never replace good geographical thinking. </li></ul><ul><ul><li>Next week- looking at the tests of significance- Mann Whitney U test and Chi Squared. </li></ul></ul>
  19. 19. Homework <ul><li>Find a definition for the following. </li></ul><ul><li>You will need this for later lessons. </li></ul><ul><ul><li>Correlation </li></ul></ul><ul><ul><li>Positive and Negative Correlation </li></ul></ul><ul><ul><li>Spearmans Rank Correlation Coefficient (with method if possible). </li></ul></ul>