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Understanding the mean 2


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Last updated 26 October 2012 15:28 by NZTecAdmin
Understanding the mean 2 (PDF, 54 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 the mean as the ‘balance point’ of a set of data. This concept describes the mean as the point on a number line where the data on either side of the point is evenly balanced. This activity builds on the “Understanding the mean 1” activity and is targeted at those learners who understand the add-up-and-divide algorithm for finding the mean but do not understand the mean as a measure of the centre of a set of data.

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 mean is affected by very large or very small values (outliers) that are outside the range of the rest of the data.
    • 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 have an understanding of the mean and how to calculate it.

Resources

  • Sticky notes.

The guided teaching and learning sequence

1.Discuss with the learners the understanding that both graphs and statistics can provide a sense of the shape of the data, including how spread out or how centred they are. Tell the learners that some statistics measure the spread of data and others measure how centred the data is.

2. Ask the learners to explain what they think the mean of a set of numbers tells you. It is likely that the learners will have a variety of understandings of the concept of a mean including some or all of:

  • adding up the numbers and dividing by how many numbers there are (while this is an accurate description of an algorithm used to find the mean, it does not describe what the mean actually is)
  • the middle of the numbers (this is an ambiguous definition as it could describe the mean, the median or the mid-point between the highest and lowest values)
  • the most common number (this is the mode, not the mean).

3. Tell the learners that this session develops an understanding of the mean as a balance point. Explain that the mean is one measure of the centre of a set of data.

4. Draw a number line on the board and label the points from 0 to 10. Place 6 sticky notes one above the other above the number 5 on the number line:

Image of a number line on the board.

Ask:

“What is the balance point value on this number line?”
“How do you know?”

The learners may only see the mean as the result of adding up all the values and dividing by the number of values. This activity aims to encourage them to see the mean as the balance point. The mean of a set of values that are all the same must be that value.

5. Ask the learners to suggest a way that the data could be changed so the balance point value remains as 5. They are likely to suggest moving two of the sticky notes in opposite directions, so that one is placed at 4 and the other at 6. This will result in a symmetrical arrangement.

Image of a symmetrical arrangement.

6. Put the sticky notes back on the 5 and ask:

“How many ways could you keep the balance point at 5 if we move one of the values to 2?”

Encourage the learners to see that a move of three spaces to the left could be balanced by moving one value three spaces to the right, or by moving three values one space to the right, or by moving one value one space to the right and one value two spaces to the right. Each time a balanced move is made you have made a new distribution with the same mean.

Image of new distributions with the same mean.
Help the learners notice that the mean only defines a ‘centre’ of a set of data and so by itself is not a very useful description of the shape of the data. The balance approach clearly illustrates that many different distributions can have the same mean.

7. As the balance concept does not lead directly to the add-up-and-divide algorithm for computing the mean, it is important to link the result of the add-up-and-divide algorithm to the ‘balance’ that was found by moving the sticky notes. Discuss how the balance point approach can be used to find the mean of a set of numbers. Provide pairs of learners with the following sample data set and ask them to model this data set on a number line with sticky notes. Costs of meals: $14, $22, $17, $18, $17, $16, $12, $12.

8. Ask the learners to predict where the mean might be. Encourage them to avoid using the add-up-and-divide algorithm, and instead have them think about the ‘balance point’ of the meal prices.

“What do you think the mean cost for these meals is?” “Why?”

9. Ask the learners to determine the actual mean by moving the sticky notes in towards the centre. For each move of a sticky note one space to the left, there must be a corresponding move one space to the right. Encourage the learners to move sticky notes multiple spaces or move multiple sticky notes at one time.

Image of the actual mean.
“What is showing when you have moved all the sticky notes to one point?” (16 is the balance point or the mean.)

Ask for volunteers to show how they moved the sticky notes to find the balance point (mean) of 16.

10. Depending on your learners’ level of understanding, you may want to pose another problem for them to solve by moving sticky notes to find the balance point (mean). Cost of meals: $21, $24, $27, $25, $25, $25, $29, $29, $30, $35 (mean = $27)

Before they move the sticky notes, ask the learners to estimate where they think the mean of the data set will be. Notice if they are estimating where the balance point or ‘centre’ of the data could be.

11. Link the ‘balance’ that was found by moving the sticky notes to the result of the add-upand- divide algorithm:

(21 + 24 + 27 + 25 + 25 + 25 + 29 + 29 + 30 + 35 = 270; 270 ÷ 10 = 27)

Although the learners may use calculators to add up the values, encourage them to do the division by 10 mentally.

12. Write the following prices on the board, explaining that they are the cost of items in a ‘lucky dip’: $8, $12, $3, $5, $7, $1. Ask:

“What is the mean cost of the items?”
(36 ÷ 6 = $6)
“What would happen to the mean cost if the $1-item were removed?” (The mean cost would increase significantly.)
“What would be the new mean cost?” ($7)
“How did you calculate it?”

Suppose that one new item were added to the original set of prices that increased the mean from $6 to $8. What would be the cost of the new item? Ask the learners to share their strategies for calculating the cost of the new item. (One solution is that you know that the total price for 7 items with a mean cost of $8 would be 7 x 8 = $56. The new item must therefore be 56 – 36 = $20.)

Follow-up activity

Ask the learners to create ‘mean’ problems for others to solve. Expect the learners to use the add-up-and-divide algorithm to compute the mean. Encourage the learners to reflect on the calculated mean, asking if the answer they have obtained is ‘reasonable’ in relation to where the ‘balance point’ of the data is.

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