Understanding Data

Class 09 Maths

An average of averages can be computed using the weighted mean. The weighted mean can also be used to calculate the mean when certain values are added to or removed from data.

The weighted average or weighted mean:

$$ x = \frac{w_1x_1 + w_2x_2 + \cdots + w_nx_n}{w_1 + w_2 + \cdots + w_n} $$

Mixtures

The weighted average is also useful in calculating the concentration of mixtures.

Example: Two glasses of equal quantities of lemonade are prepared. One glass has 10% jaggery and the other has 20% jaggery. If we mix the lemonade from both glasses, what is the concentration of jaggery in the mixture?

Since the glasses had equal quantities of lemonade, the concentration of jaggery in the combined mixture is midway between the concentrations of jaggery in the individual glasses.

Example: A bowl of 500 mL lemonade has 10% jaggery. A glass of 200 mL lemonade has 20% jaggery. If we mix the lemonade from the bowl and the glass, what is the concentration of jaggery in the mixture?

Since the bowl has a larger quantity of lemonade, the concentration of jaggery in the mixture will be closer to 10% (less than midway).

Example: Brass is an alloy of copper and zinc. Batch A of brass weighs 200 kg of which 70% is copper. Batch B weighs 120 kg of which 50% is copper. Batch C is 45% copper. When all three batches are combined, we get an alloy that is 55% copper. What is the weight of Batch C?

$$ \frac{200(70) + 120(50) + 45x}{200 + 120 + x} = 55 $$

$$ \frac{14000 + 6000 + 45x}{320 + x} = 55 $$

$$ 20000 + 45x = 17600 + 55x $$

$$ 2400 = 10x $$

$$ \boxed{x = 240 \text{ kg}} $$

Custom Weights

Whenever different parts of data have different number of values or levels of importance, the weighted mean can be used by assigning appropriate weights to each part of the data. Examples include rating systems and combining marks of different assessments to get a final score.

Example: Rehmat’s marks in Maths are as follows: 60% in internal tests, 64% in the project, 73% in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3 : 2 : 5. What is Rehmat’s annual score in Maths?

$$ \text{Annual Score} = \frac{3(60) + 2(64) + 5(73)}{3+2+5} $$

$$ = \frac{180 + 128 + 365}{10} $$

$$ \boxed{\text{Annual Score} = 67.3\%} $$

Visualising and Interpreting Data

Stacked bar charts are used when we want to compare the totals but also want to keep track of the components that make up the totals.

Stacked bar charts may be used in place of cluster bar charts if the total of each cluster is also a matter of interest.

100% stacked bar charts, like pie charts, compare proportions and not absolute values. It is similar to a pie chart - the complete angle in a pie chart is split proportionately based on the quantities, whereas here, a fixed length is split proportionately.

Given a stacked bar chart, a corresponding 100% stacked bar chart can be made but not vice versa.