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Understanding the mean 1


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Last updated 26 October 2012 15:28 by NZTecAdmin
Understanding the mean 1 (PDF, 45 KB)

Analysing Data for Interpretation progression, 6th step

The purpose of the activity

In this activity, the learners develop an understanding of the concept of a mean as a number that represents what all the data items would be if they were levelled out to be the same. This activity is targeted to those learners who have no understanding of the mean or those who know the add-up-and-divide algorithm for the mean but do not understand why it works.

The teaching points

  • The learners understand that measures that describe data with numbers are called statistics.
  • The learners understand that both graphs and statistics can provide a sense of the shape of the data, including how spread out or how centred they are. Having a sense of the shape of the data is like having a big picture of the data rather than just having a collection of numbers.
  • The learners understand that an average is a single number that is descriptive of a larger set of numbers. The mean, median and mode are specific types of average. Averages are measures of how centred the data is or the central tendency of the data.
    • The mean is computed by adding all of the values in the set and dividing the sum by the number of values added.
    • The median is the middle value of an ordered set of data. The median is relatively easy to compute and is not affected (like the mean is) by one or two very large or very small values that are outside the range of the rest of the data.
    • The mode is the number or value that occurs most frequently in the data. This statistic is least useful as often the mode does not give a very good description of the set. For example, 9 is the mode in the following set of values: 1, 1, 2, 2, 3, 4, 9, 9, 9.
  • Discuss with the learners relevant or authentic situations where it is necessary to understand ‘mean’ and how to calculate it.

Resources

  • Cubes (preferably ones that link together
    • these are available from educational resource centres).
  • Paper strips (about 40 centimetres long).
  • Sticky tape.

The guided teaching and learning sequence

1. Have each learner cut a strip of paper to the length of their foot. Ask them to record their names and the length of their foot in centimetres on the strip. Put the learners into groups of four or eight and give each group a roll of sticky tape. The task is for each group to come up with a method for finding the mean (the typical length) without referring to the amounts written on the strips. Pose the question:

“How can you use the strips of paper to find out the mean length of feet in your group? (You must not look at the length written on the strips.)"

2. If the learners do not suggest a solution, prompt them by saying that the mean length is the length if all the lengths were the same. It may help to describe the mean as ‘the typical length’. One option is to join the foot-strips end to end and then fold the combined strip into equal parts so that there are as many sections as learners in the group. Another option is to cut one piece from each of the larger foot strips and attach it to the end of each smaller piece until all the pieces are the same length.

The learners can then measure the length of any one piece.

3. Ask the groups to share their approaches. Ensure that the learners understand that the length of the ‘levelled’ foot strip is the mean (typical length) of the foot lengths.

4. Next pose the question:

“How can we find the mean for the whole class?”

(Although it is possible to join the strips into a single length (say for 15 learners), it isn’t practical to actually fold the strip into 15 equal parts.)

5. Ask the learners to think about how they could calculate the mean foot length without folding the long strip: “Without folding the strip, how can we find the mean (typical) foot length?” (The total length of the strip is the sum of the 15 individual foot strips. To find the length of one section, if the strip were actually folded in 15 parts, you divide by 15.)

6. Discuss with the learners how this process illustrates the usual add-up-and-divide algorithm for finding the mean.

7. Write the following meal prices for a group of 10 people on the board and ask the learners what they think everyone would have to pay if the group were to share the account.

$12, $17, $17, $18, $19, $20, $22, $22, $23, $35

8. Ask:

“What do you think the mean meal price per person would be?” “Why do you think that?”

Listen for responses that confirm that the learners have understood the concept of ‘levelling’ to find the mean. In this case, the learners might see that $10 could be taken from $35 and put onto $12 to make $25 and $22 respectively. It then seems reasonable for the mean price to be around $20 or $21.

9. Next ask the learners to use the add-up-and divide method to work out the mean price. Ask:

“If the restaurant will only allow one account per table, exactly how much will each person pay if they all agree to pay the same amount?” (12 + 17 + 17 + 18 + 19 + 20 + 22 + 22 + 23 + 35 = 205; 205 ÷ 10 = $20.50)

Note: Although the learners may use a calculator to sum the amounts, encourage them to mentally divide the total by 10.

10. Depending on your learners’ level of understanding you may want to work through one or two more examples as a class before asking them to solve problems independently. Possible examples for further guided learning or independent work are:

  • mean meal cost: $16, $16, $19, $19, $21, $22, $25, $26, $28
  • mean daily temperature for the week: 16°, 17°, 23°, 18°, 18°, 19°, 20°.

Before the learners calculate the mean in each instance, encourage them to apply the ‘levelling’ approach to obtain an estimate of the mean. Alternatively ask them to reflect on the calculated mean, asking if the answer they have obtained is ‘reasonable’.

Follow-up activity

Pose problems that require the learners to find the mean of a set of values. Examples could include mean temperature, mean test score and mean height.

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