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Arithmetic mean, median and mode

الأسبوع 9 · الوسط الحسابي والوسيط والمنوال: من القوائم والجداول التكرارية والبيانات المضللة · 3 دروس
3

Median and mode: choosing the best measure, and misleading data

Mathematics Prep 1 Term 1, Unit 3, Lesson 2
GoalFind the median and mode, see how a new value changes them and choose the best measure.
New words
median · the middle value of ordered datamode · the most frequent value; there may be none or severalextreme value · a value much bigger or smaller than the otherstypical · a value that represents most of the data
Explanation

Mean, median and mode are three ways to describe the centre of data. To find the median, put the data in order first. For an even number of values take the mean of the two middle ones. The mode is the most frequent value; a set can have no mode or more than one. The mean is pulled by very large or very small values, so with an extreme value the median is often more typical. For things that are counted in categories, such as the most popular shoe size, use the mode. When a new value is added, the mean usually changes, the median may change a little, and the mode may stay the same.

★Remember
Order the data before finding the median. An extreme value pulls the mean. The mode can be none or several.
Worked examples
A new value. Pupils present in five days: 28, 30, 29, 30, 28. Mean = 145 ÷ 5 = 29. Median = 29 (in order 28, 28, 29, 30, 30). Modes: 28 and 30. A sixth day has 35 pupils. New mean = 180 ÷ 6 = 30. New median = (29 + 30) ÷ 2 = 29.5. The modes are still 28 and 30.
MeasureValue
mean900 LE
median430 LE
modenone
A misleading average. Five weekly pay amounts at a bakery are 400, 420, 430, 450 and 2,800 LE. The mean is 4,500 ÷ 5 = 900 LE, but four of the five workers earn about 400 to 450 LE. The median, 430 LE, is more typical.
Best measure. A shoe shop sold these sizes: 36, 37, 37, 38, 37, 39, 40, 37, 38. The shop owner wants to know which size to order most, so the mode helps: 37 occurs 4 times, more than any other size.
Class activity

Which average? With a grown-up, collect the ages of eight people in the family or street, including one much older person. Work out the mean and the median. Which describes the group better? Why?

Practice
1
Find the median and the mode of 3, 5, 5, 7, 9, 5, 8.
2
The data 4, 6, 8, 10 get a new value 40. Find the mean and the median before and after.
3
Which measure tells a shop the most popular size: mean, median or mode?
4
Find the mean and median of 2, 3, 3, 4, 100. Which is more typical?
5
Salma says every set of data has exactly one mode. Is she right?