NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 05 Tales by Dots and Lines PDF Download

Chapter 5: Tales by Dots and Lines

5.1 The Balancing Act

Last year, we learnt about the mean and median. Recall that the mean of some data is the sum of all the values divided by the number of values in the data. The median is the middle value when the data is sorted.

We shall try to understand the mean and median from a different perspective and see how the mean behaves with changing data.

Consider any 2 numbers. Find their average/arithmetic mean. Repeat this by taking other pairs. What do you observe?

For example, let the two numbers be 3 and 7. Their average is \( \frac{3 + 7}{2} = 5 \). Taking another pair of numbers, say 8 and 9, their average is \( \frac{8 + 9}{2} = 8.5 \). Visualising these as dot plots we get

[Figure: Dot plots showing numbers 3 and 7 with mean at 5, and numbers 8 and 9 with mean at 8.5, See in your textbook]

Notice that the mean is exactly halfway between the two numbers.

We have learnt earlier that the arithmetic mean is a measure of central tendency and represents the 'centre' of the data. Let us see how the mean represents the 'centre' in the case of 3 numbers.

Calculate and mark the mean of each collection of data below.

[Figure: Four dot plots with data collections to mark means, See in your textbook]

Teacher's Note

When you calculate the mean, remember that you are finding the sum of all the values and dividing by how many values there are. For example, if the data is 6, 7, 7, 8, the mean is \( \frac{6 + 7 + 7 + 8}{4} = \frac{28}{4} = 7 \). Once you have the mean, mark it on the dot plot to see where the balance point is.

Can you explain how the mean is the centre of each collection?

Mark the mean for the collections below.

[Figure: Four dot plots with data collections, See in your textbook]

Can you explain how the mean is the centre of each collection?

Is the mean the midpoint of the two endpoints/extremes of the data? It is not always so. Instead, the total distances are equal on both the sides of the mean. This is illustrated through the following dot plots.

[Figure: Four dot plots showing distances from mean on left-hand side (LHS) and right-hand side (RHS), See in your textbook]

Verify that this holds for all the collections of data shown earlier.

Can there be more than one such 'centre'? In other words, is there any other value such that the sum of the distances to the values lower than it and the values higher than it will still be equal?

In the case of the collection 10, 10, 11, and 17 whose mean is 12, suppose there is a different centre larger than 12.

Clearly, all the distances on the LHS will increase and the distances on the RHS will decrease. Thus, it is no longer the 'centre'. Similarly, for any value smaller than 12, the distances on the LHS will decrease while those on the RHS will increase.

Both these cases are illustrated in the following diagram. Therefore, there is only one centre.

[Figure: Two dot plots showing why other values cannot be the centre, See in your textbook]

Will including a new value in the data increase or decrease the mean?

When a new value greater than the mean is included, the mean increases to maintain the balance between the sum of distances on the LHS and RHS, as illustrated below.

[Figure: Three dot plots showing how adding a value greater than the mean increases the mean, See in your textbook]

Similarly, if a value smaller than the mean is included, then the new mean will be less than before.

Teacher's Note

Think of the mean as a balance point on a number line. If you add a value much bigger than the current mean, the balance point shifts right, so the mean goes up. If you add a value much smaller than the mean, the balance point shifts left, so the mean goes down. If you add a value equal to the mean itself, the balance stays the same and the mean does not change.

What happens to the mean when an existing value is removed? When will the mean increase, decrease, or stay the same?

What happens to the mean if a value equal to the mean is included or removed?

Try to explain this using the fair-share interpretation of mean that we studied last year.

Unchanging Mean!

We saw earlier how the mean varies when a value is included or removed.

Key Points

• The mean is the balancing point of data: the sum of distances from the mean to values on one side equals the sum of distances to values on the other side.
• The mean is not always the midpoint of the smallest and largest values, but it is the unique centre where distances balance.
• Including a value larger than the mean will increase the mean, while including a value smaller than the mean will decrease it.
• If you add or remove a value that equals the mean, the mean stays the same.

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