SCATTERPLOTS. We can talk about the correlation or relationship or association between two variables and mean the same thing.

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1 SCATTERPLOTS When we want to know if there is some sort of relationship between 2 numerical variables, we can use a scatterplot. It gives a visual display of the relationship between the 2 variables. Graphing a scatterplot: 1. Decide which is the EV & RV 2. Explanatory variable = horizontal axis 3. Response variable = vertical axis 4. Make sure scales are appropriate & accurate 3D Interpreting Scatterplots We can talk about the correlation or relationship or association between two variables and mean the same thing. Step 1: Look to see if there is a clear pattern. If so, proceed to Step 2. Step 2: Look for DIRECTION (or Polarity) i.e. whether it is positive or negative. Positive Relationship Negative Relationship Scattered randomly or No Association Step 3: Make a judgement about the STRENGTH of the relationship between the variables. This is based on the spread of the points. Strong Moderate Weak Strong Moderate Weak

2 Step 4: Observe whether the pattern of the points appears to be LINEAR (in a straight line) or not. Can also be referred to as linearity or form. Linear: Non Linear: Step 4: Identify and investigate any outliers. Sometimes they are a mistake. Sometimes they are genuinely extraordinary data and should be included. EXAMPLES 1. The relationship between the variables x and y is shown on the scatterplot. The correlation between x and y would be best described as: A a weak positive association B a weak negative association C a strong positive association D a strong negative association E non-existent 2. An investigation is made into the number of freckles on the back of a hand and the age of the subject. A strong association was found to exist. In this investigation, age is the explanatory variable and the number of freckles is the response variable. You would expect the association to be: A negative B positive C bivariate D weak E categorical

3 Pearson s Product - Moment Correlation Coefficient Also called Correlation Coefficient or r. This is a more precise tool. In summary it is a measure of the tendency of data to lie on a straight line. The value of r ranges from -1 to 1 i.e. -1 r 1. Following are examples of scatterplots: The value of the Correlation Coefficient indicates the strength of the linear relationship between 2 variables. The following diagram gives a guide to the strength of the correlation based on the value of r.

4 3. For each example below, describe any associations in terms of strength, direction and form. Identify any outliers where appropriate. Estimate the value of r. Refer to 3E interactive 3E

5 4. A set of data relating the variables x and y is found to have an r value of The scatterplot that could represent this data set is: A B C D E 3F A formula can be used to calculate r : r = 1 n 1!!!! x! x s! y! y s! OR alternatively, using your calculator: Calc Lin Reg 2 variable Only use r when data is: Numeric Linear Has No outliers (check by creating a graph) Coefficient of Determination The degree to which one variable can be predicted from another linearly related variable is given by the Coefficient of Determination and is calculated by squaring the Correlation Coefficient, therefore Coefficient of determination = r 2. Values range from 0 to 1 i.e. 0 r 2 1. NB: Answer normally expressed as a percentage i.e. r 2 = 0.58, therefore Coefficient of Determination is 58%. NB: When calculating r from r 2, there can be 2 possible answers, positive or negative. information is given, both answers must be given. Always check for a graph or equation. 5. Calculate r given r 2 = If no further Interpreting r 2 The Coefficient of Determination tells us that (r 2 100%) of the variation in the RESPONSE variable is explained by the variation in the EXPLANATORY variable.

6 6. The coefficient of determination for a set of data relating age and pulse rate is 0.7. NB: Coefficient of determination > 30% can be regarded as having significant predictive power. Correlation and Causality 3G The value of r was calculated for the following 2 variables: height of a footballer & number of marks he takes. It was 0.86 which entitles us to say there is a strong association between the variables. We CANNOT, however, assert that the height of a footballer causes him to take a lot of marks. Being tall might assist in taking marks but there will be many other factors which come into play, for example skill level, accuracy of delivery, etc. A correlation tells you about the strength of the association between two variables but no more. It tells you nothing about the source or cause of the association. To establish causality, you need to conduct an experiment where the explanatory variable is deliberately manipulated and the response variable is kept constant/controlled. Non-causal explanations for an association 1. Common response when two variables are associated because they are both strongly associated with a common third variable. E.g. Strong positive association between number of people using sunscreen and number of people fainting. They are associated because of temperature. 2. Confounding variables when there are at least 2 possible causal explanations for the observed association and no way of disentangling their separate effects, they are said to be confounding. This is because there is no way of knowing which is the actual cause of the association. E.g. Crime rates & unemployment are strongly correlated, however, economy could have caused the problem. 3. Coincidence occasionally, it is impossible to identify any feasible confounding variables to explain a particular association. In these cases, we often conclude that the association is spurious and it has happened by chance. E.g. strong correlation between consuming margarine & divorce rate in Maine, USA. Association / correlation and causation By itself, an observed association between two variables is never enough to justify the conclusion that two variables are causally related, no matter how obvious the causal explanation may appear to be. Which Graph? 3H 3I