1. STATISTICS Pearson’s Product Moment Advanced Higher

2. STATISTICS Pearson’s Product Moment Finding a relationship between two variables Lesson Objectives 1. Explain why it is used. 2. List the advantages and disadvantages. 3. Understand how to apply the statistical test. 4. Apply it to a relevant context.

3. Advanced Higher STATISTICS Pearson’s Product Moment The situation A group of four students have collected primary fieldwork data from an upper course river. They are investigating whether there is a relationship between the length and width of the pebbles that they have collected from one site. What should they do? What data should they have collected to complete this investigation? How much data should they collect? How can they make sure that the data is reliable? What initial data representation skill could they utilise to discover an initial impression? What statistical test should they use to confidently state there is or is not a relationship?

4. Pebble length (cm) 0 0 Pebble width (cm) Pebble length (cam) 0 Pebble width (cm) 0 0 0 Pebble length (cm) Pebble width (cm) Positive correlation r s = +1 No correlation r s = 0 Negative correlation r s = -1 Pearson’s Product Moment -Is there a relationship? - Is it a positive relationship? - Or a negative relationship? - Or do they show nothing in common at all?

5. Null Hypothesis: There is no significant correlation between bedload length and width Alternative Hypothesis: There is a significant correlation between bedload length and width xyx - xy - y( x-x ) 2 ( y-y ) 2 (x-x)(y-y) 8.19.20.370.14 7.43.7-0.330.11 7.66.4-0.130.02 8.08.20.270.07 8.015.00.270.07 7.112.6-0.630.40 8.03.10.27.07 7.89.50.070.00 7.87.90.070.00 8.02.30.270.07 8.04.30.270.07 7.015.0-0.730.53 X = 7.73y = 5.66 TESTING THE RELATIONSHIP CL = 95% Critical Value = 0.591 R s = 0.89, therefore reject Null Hypothesis

6. There is a very weak negative correlation, as one increases the other decreases. The null hypothesis cannot be rejected as it is too close to ‘0’. The result is not statistically significant at 95% or 99%. The data set used is very small (12 is the minimum) The data set comes from one river point only. The students who collected the data may have made errors. The river may not be representative of all rivers. The point in the river may have been a plunge pool. PPM uses actual values so is a very powerful statistic PPM recognises differences that SRC would ignore as it uses the values themselves. The data is interval data and is normally distributed. There is no significant relationship between pebble length and pebble width Pearson’s Product Moment State the answer in terms of the null hypothesis. Justify the suitability of using PPM correlation coefficient. Describe how you might have made the investigation more reliable. Is the relationship statistically significant? Why not? Collect data down the whole length of the river. Use callipers to measure the pebble size and ensure accuracy. Measure a second river to ensure a wide representation of rivers were studied. Change in the weather may have meant that the data was collected on a very wet day.