Student Reasoning about Average

 

Developing Critical Numeracy Across the Curriculum

Student Reasoning about Average

Average is a term which has common meanings such as "That's average" (meaning not very good), as well as mathematical meanings such as mean, mode and median. The news often reports average ambiguously where the use of mean, mode or median is not clear.

 

Learning Sequence on Average with teacher comments

Mean, mode and median are different ways to express the central tendency of a data set. Each has particular usefulness depending on the type of data and its variation. For example:

  • Mode (most frequent) is often used when reporting categorical data. "The average man drinks beer."
  • Median describes the middle position of a data set that has been ordered from smallest to largest. It is useful in giving a sense of a central value when data sets have a few high numbers which could skew results, for example in providing average house prices.
  • Mean is a formulaic approach to analysing a data set of numbers. It uses procedures of addition and division. "The average family has 2.3 kids."

Research has found that students' concept of average usually starts with a notion of mode which is a more intuitive concept - what has the most? It then progresses to median - what is the middle value? The concept of mean requires an understanding of formal maths and calculation and is developed quite a bit later. Younger students might have difficulty in understanding what an average of 2.3 kids actually means - "Um, does it mean there are two older children and then one under ten or something?"

Development of thinking

Looking at the central tendencies of a data set can be problematic without also considering the variation. Students can get an intuitive sense of variation through graphing and comparison of different data sets within the same context.

Over the primary and middle school years it is likely that student understanding of average will develop in the following sequence, encompassing the three conventional definitions of mean, median, and mode.

  • Single colloquial expressions: okay, normal, common, the same as everyone else
  • Expressions reflecting “part” of a set:
     “the same as most people”
     “the most common result on a test”
  • Expressions reflecting a measure:
    “not really good and not really bad”
    “halfway in a set of something”
    “in the middle like most people”
    “add up some numbers and divide by how many”
  • Expression recognising that an average represents a data set:
    “an average tells us what is typical of a data, e.g., the mean is the balance point for all data values”
Average Student - Grade 7 class

Once students have the procedure for finding the mean, they develop the ability to solve problems using it.

  • Working a mean value backward knowing the number of data values, can produce the total of all values and sometimes missing data values.
  • Working with weighted means can combine means for different sized sets.
  • Means can be used to compare sets of different sample sizes.

Eventually students will have an intuition for when it is appropriate to use the mean to answer questions about data sets and will consider the variation present as well as the single mean value.