The mean

The mean The World Stone Skimming Championships take place on Easdale Island in Scotland, UK. Here are some distances in meters for each category. men women boys girls 54 36 33 18 52 28 15 46 23 14 27 26 13 Teacher notes Students will already have met the mean in middle school, so this will be more of a review for them. How would you compare each of the categories?

The mean The mean is the most commonly used average or measure of central tendency. To calculate the mean of a set of values we add together the values and divide by the total number of values. sum of values mean = number of values The mean is often denoted using the notation: . x This is read as “x bar.”

The mean men women boys girls 54 36 33 18 52 28 15 46 23 14 27 26 13 The mean distance (in meters) for each category are recorded in the table below. men women boys girls mean 44.75 m 27.50 m 28.75 m 15 m Complete this table of mean distances. Teacher notes Discuss an appropriate level of accuracy. When calculating the mean values for each of the four groups, students must remember to complete the table with the appropriate units (meters) to keep in the context of the question. To complete the second table, students must find the mean of two means; the men and women to make the Seniors, and the boys and girls to make the Juniors. Again, students must remember the correct units in the context. Photo credit: © Stanislav Komogorov, Shutterstock.com 2012 seniors juniors mean 36.13 m 21.88 m Calculate means for the Seniors and the Juniors (with male and female results combined).

Outliers and the mean Here are the results (in m) for one stone-skimming contestant. 36.6 36.2 35.3 34.7 37.3 34.4 15.9 36.3 Discuss: Are there any outliers? Why might this outlier have occurred? Will the outlier increased or decrease the mean? Calculate the mean with and without with the outlier. How much does it change? Teacher notes 15.9 is an outlier. Any mistake made by the contestant may have caused it e.g. a slip of the hand or foot when throwing. This outlier will decrease the mean. The mean with the outlier is 266.7 ÷ 8 = 33.3 m. The mean without the outlier is 250.8 ÷ 7 = 35.8 m. The outlier affects the mean by 2.5 m. When conducting an experiment, we need to be able to draw accurate conclusions from the data. If there is an outlier in the data, especially one caused by human error e.g. a slip of the hand, this will make the results less accurate, so it is may be more appropriate to remove any outliers. For more about outliers and a mathematical definition of it, see the presentation The range and interquartile range.ppt. Mathematical Practices 4) Model with mathematics. Students should see that the outlier affects the model (the mean) so that it is less useful for basing predictions on. They should consider these things when choosing their model. Leaving out the outlier when finding the mean gives a more useful model for the data set. It may be appropriate in research or experiments to remove an outlier before carrying out analysis of results. Why do you think this is?