Agriculture in Education: an educational resource for the Year 8 Mathematics Statistics and Probability – Data representation and interpretation Resource: Dairy Herd Data Content Descriptor: Investigate the effect of individual data values, including outliers, on the mean and median (ACMSP207) Source: Australian Curriculum, Assessment and Reporting Authority (ACARA), downloaded from the Australian Curriculum website on 18 December 2014 Learning Outcome: Students explain issues related to the collection of data and the effect of outliers on means and medians in that data. Description: This resource provides a context to engage students in an investigation of data including exercises to find the mean and median and the effect of outliers on means and medians in that data. The context is that of three farmers trying to increase the milk yield of their dairy herds. Students will investigate the yield data for individual cows in the herds and analyse the effect of increasing feed to these cows. From this data they will draw conclusions and make suggestions about how the farmers can increase their yields most economically. They will also look to investigate how combining herds would benefit or disadvantage the farmers. Page 1 Setting the scene: This resource is ideal to include as an activity alongside an agriculture program such as Cows Create Careers. It will also work well standalone, however as a way to make the resource more relevant and to engage students, teachers might consider showing one of the following YouTube videos that are amusing parodies with positive messages about farming. “Feedin’A Nation”- Dairy Farming parody song “CHORE” (Roar Parody) by the Peterson Brothers Explain to students that they are each going to support a dairy farmer who owns a small herd of cows. The farmer has been trying to find a way to improve the output or yield of milk that his cows produce and has been keeping a record of the yield that each cow produces. Work Task 1: Means, Medians and Outliers Each student should be given a worksheet with the two sets of data (data sets 1 and 2) and instructions to help them help the farmer. This task involves calculating means, medians, and the effect of outliers on a data set and comparison of two linked data sets. Students are asked to comment on the validity of the data and interpret it for a simulated real world scenario, thereby demonstrating their understanding of what the data represents. Work Task 2: Extension Activity – Interpreting Data Explain to students that three farmers have decided to see if by working together they can do things more economically. It may be beneficial for students to work in pairs or groups of three for this task in order to promote broader ideas and discussion. Provide students with the worksheet for this task, explaining that it provides a summary of the data for the two other farms. The worksheet invites students to combine data and analyse the effects of the additional data. Further exercises related to outliers, means and medians will challenge and scaffold student. Assessment: Page 2 The student answers on these worksheets will serve as very strong evidence for the learning outcomes. Teacher support resources For further ideas, Agriculture in the Classroom website, although an American website, has some wonderful resources. Also see http://www.growtolearn.org/view/schoolgarden101 Page 3 Worksheet – Means, Medians and Outliers Work Task 1: Farmer Jones has been recording how much milk each cow has been producing each day over a typical week and has given you the data in a table: Data Set 1 1 2 3 4 5 6 7 Daisy 29 28 30 32 30 31 29 Helga 33 29 30 30 31 30 32 Rosy 25 28 27 29 26 27 29 Gina 31 30 32 28 28 27 30 Beryl 28 27 29 26 28 27 29 Gertie 30 30 31 30 32 28 28 Tessa 28 25 24 26 28 25 24 Trish 30 27 33 35 30 29 28 Ethel 28 28 27 30 31 32 31 Clarice 27 29 30 28 25 24 26 Flo 31 32 31 32 30 31 29 Martha 38 37 38 39 36 37 38 Shirl 27 30 31 29 30 27 33 Mildred 26 29 26 28 27 29 26 Lilly 30 32 30 31 29 30 27 Doris 30 30 31 30 32 28 28 Barbara 27 29 26 27 29 27 29 Lazy Lou 10 12 11 10 10 12 9 Page 4 Total Mean Median 1 2 3 4 5 6 7 Minnie 29 26 28 27 29 26 27 Bessie 31 30 32 28 28 27 30 Star 48 46 45 45 43 44 49 Maude 26 27 29 30 28 25 24 Betty 16 18 15 14 17 17 15 Nancy 22 24 22 20 23 25 22 Judy 36 35 33 35 31 36 33 Total Mean Median What you need to know: The mean and median are measures of the centre of a set of data. The Mean is what we commonly refer to as the “average” of a set of numbers, where you add up all the numbers and then divide by the number of numbers. The Median is the “middle” value in the list of numbers. To find the median, your data must be first listed in numerical order. If there are “x” numbers in your data then count x /2 from each end to find the middle number(s). If there is an odd number of data, x/2 will give you a number including a half value. In this case round up to the next whole number. If there is an even number of data when you count x /2 from both ends you will find two “middle” numbers, in which case you will need to add the two middle numbers together and divide by two to find the median. Example a) 3, 6, 7, 9, 10, 16, 18 There are 7 numbers in this set, so the middle number is the fourth number, which makes the median 9 in this case. Example b) 2, 4, 5, 9, 11, 12, 15, 17, 18, 22. There are 10 numbers in this set so the two middle numbers are 11 and 12. Therefore the median is (11 plus 12) divided by 2 equals 11.5 Page 5 An Outlier in a set of data is a number that is at the extreme end of the range of numbers, example consider the set [66, 72, 74, 21, 67, 73,101, 69]. The outliers in this set are 21 and 101. Now answer the following questions. Include your calculations where applicable. 1. The farmer would like to increase the yield of the cows but needs your help with analysing the data. Complete the table above with weekly total yields for each cow, the mean daily yields and the median daily yields. 2. What is the mean weekly yield per cow for the herd? 3. What is the median weekly yield? 4. There are two cows whose yield is either exceptionally bad or good. What two cows are they? 5. The data for these two cows are the outliers for this set of data. If we remove the outlier data from our herd’s data how does it affect the mean and median of the weekly yield per cow of the herd? 6. The farmer wants to know what the “average” cow yield is in order to judge the cow’s production. Should we use the mean and median including the outliers or without the outliers? Explain why this is important. The farmer up to this stage has always fed the cows exactly the same amount of feed. The farmer is wondering if their feed is increased whether their milk yield will increase at the same rate. The farmer decides to run a trial where all the cows are fed an extra 10 percent more feed. The data showing the milk yield each for the trial is in “data set 2” Page 6 Data Set 2 1 2 3 4 5 6 7 32 31 33 35 33 34 32 Helga 35 31 32 32 33 32 34 Rosy 28 31 30 32 29 30 32 Gina 34 33 35 31 31 30 33 Beryl 29 28 30 27 29 28 30 Gertie 32 32 33 32 34 30 30 Tessa 32 29 28 30 32 29 28 Trish 33 30 36 38 33 32 31 Ethel 31 31 30 33 34 35 34 Clarice 27 29 30 28 25 24 26 Flo 34 35 34 35 33 34 32 Martha 39 38 39 40 37 38 39 Shirl 30 33 34 32 33 30 36 Mildred 29 32 29 31 30 32 29 Lilly 33 35 33 34 32 33 30 Doris 33 33 34 33 35 31 31 Barbara 30 32 29 30 32 30 32 Lazy Lou 15 17 16 15 15 17 14 Minnie 32 29 31 30 32 29 30 Daisy Page 7 Total Mean Median Percentage Difference 1 2 3 4 5 6 7 Bessie 33 32 34 30 30 29 32 Star 48 46 45 45 43 44 49 Maude 29 30 32 33 31 28 27 Betty 20 22 19 18 21 21 19 Nancy 26 28 26 24 27 29 26 Judy 37 36 34 36 32 37 34 Total Mean Median Percentage Difference “Average Cow” Now you are to compare the two sets of data that you now have. (You will need to calculate totals, means and medians for the new data). 7. What is the difference in terms of percentage increase or decrease of the milk yield for each cow? 8. Which cow best represents the “average” cow? Is it more valid to refer to the Mean or the Median cow in this case? Why? 9. What does this information lead you to think about the effect of increasing the feed for the “average” cow? 10. How reliable do you think the farmer’s data would be? What could the farmer do to make the data more accurate? 11. What advice will you give to the farmer? Work Task 2: Worksheet > Extension Activity – Interpreting Data Three farmers have decided to see if by working together they can do things more economically. You have been advising Farmer Jones. Farmer Brown and Farmer Jenkins have produced similar data for their dairy farms and they have brought along summaries to show you. Page 8 Farmer Brown Has 30 cows. The mean milk production is 28 litres per cow per day. The median cow yield is 203 litres per week. There are two outliers in the herd data. Their mean daily milk yields are 13 and 14 litres respectively. Farmer Jenkins Has 36 cows. The mean milk production is 32 litres per cow The median cow yield is 270 litres per week. There are two outliers in the herd data. Their mean daily milk yields are 16 and 44 respectively. Answer the following questions showing any calculations needed. 1. What is the total milk production for a week for each farmer’s herd? 2. Farmer Brown and Farmer Jenkins both included the outliers in their calculations of the mean and medians. By removing the outliers what would be the mean weekly yield for Farmer Jenkin’s herd? 3. What effect would there be on the median number for a week’s yield in Farmer Jenkin’s herd if we remove the outliers? 4. By the combining the herds do the farmers gain anything in terms of the new herd daily yield per cow? Who would gain the most by combining herds? Page 9 AgriFood Skills Australia General inquiries: Phone: zero two six one six three seven two zero zero Fax: zero two six one six two zero six one zero Email: mailto:reception@agrifoodskills.net.au Web: http://www.agrifoodskills.net.au Location Level Three, 10 to 12 Brisbane Avenue Barton Australian Capital Territory 2600 Postal address Post Office Box 5450 Kingston Australian Capital Territory, two six zero four Developed by John Tucker care of AgriFood Skills Australia © AgriFood Skills Australia 2015 This work is licensed under a Creative Commons Attribution – Share Alike 3.0 Australia Licence. Page 10