Class 9 › Mathematics › Chapter
How Quantities Combine: Understanding Data
Average of averages, mixtures, the weighted mean and custom weights, updating a mean when values are added or removed, and reading cluster, stacked and 100% stacked bar charts.
1. Sum, Mean and Average of Averages
Key relation
$$\text{Sum} = \text{Mean} \times (\text{Number of values})$$
- Why it matters: mean = sum $\div$ number of values, so a known mean and size give back the total. Totals can be added directly; averages cannot.
- Wrong shortcut: adding two group averages and dividing by 2 treats both groups as equally important. That is correct only when both groups have the same number of values. A larger group must count for more.
- Correct method: (1) turn each average back into a total (average $\times$ size), (2) add the totals, (3) divide by the total number of values.
Average of two collections
$$\dfrac{an + bm}{n + m}$$
Average of three collections
$$\dfrac{ap + bq + cr}{p + q + r}$$
Here
$a, b, c$ are the means and $n, m$ (or $p, q, r$) are the numbers of values in the collections.
The combined average is pulled away from the midpoint toward the mean of the collection with more values.
Derivation
1. Collection 1 has $n$ values $x_1, \dots, x_n$ with mean $a$, so $x_1 + \cdots + x_n = an$.
2. Collection 2 has $m$ values $y_1, \dots, y_m$ with mean $b$, so $y_1 + \cdots + y_m = bm$.
3. Combined mean $= \dfrac{\text{sum of Collection 1} + \text{sum of Collection 2}}{n + m} = \dfrac{an + bm}{n + m}$.
- Equal sizes (special case): if every collection has the same size $p$, then $p$ cancels: $\dfrac{ap + bp + cp}{3p} = \dfrac{a + b + c}{3}$. So the plain average of the averages is correct only when the sizes are equal.
- Where the answer lies: the combined mean is always between $a$ and $b$, and closer to the mean of the larger collection.
In simple words: a group average is only a summary of a group. To merge groups, rebuild each group's total from its average and size, add the totals and divide by the total count.
2. Mixtures and Concentration
Concentration: the part of a mixture that is one particular substance, written as a fraction, decimal or percentage.
Concentration
$$\text{Concentration} = \dfrac{\text{Quantity of the substance}}{\text{Quantity of the mixture}}$$
Concentration of a mixture of $n$ parts
$$x = \dfrac{w_1x_1 + w_2x_2 + \cdots + w_nx_n}{w_1 + w_2 + \cdots + w_n}$$
where $x_1, x_2, \dots$ are the concentrations and $w_1, w_2, \dots$ are the quantities (volumes or weights) mixed.
The shade of the mixture depends on how much of each liquid is poured in, not only on how strong each one is.
- Substance in the mixture = sum of (quantity $\times$ concentration) of every part; quantity of the mixture = sum of all the quantities.
- Equal quantities: the result is exactly midway between the concentrations.
- Unequal quantities: the result is closer to the concentration of the larger quantity. In general, each concentration influences the result in proportion to the quantity it contributes.
- Estimate first: the result always lies between the smallest and the largest concentration. Use this to check the answer.
- Pure solvent has concentration $0$: adding plain water only dilutes a solution.
- Dilution: the amount of the substance does not change when water is added, so (old concentration $\times$ old quantity) = (new concentration $\times$ new quantity).
- Different substances in one mixture: find each substance's concentration separately. The denominator is always the total mixture, not the solution that substance came from.
- Units: measure all quantities in the same unit before combining.
Finding an unknown quantity
1. Let the unknown quantity be $y$.
2. Write the substance in each part as quantity $\times$ concentration (use $y \times$ concentration for the unknown part).
3. Form: $\dfrac{\text{total substance}}{\text{total quantity}} = \text{given concentration}$.
4. Multiply both sides by the total quantity and solve the linear equation for $y$.
3. Weighted Mean
Weighted mean (weighted average)
$$x = \dfrac{w_1x_1 + w_2x_2 + w_3x_3 + \cdots + w_nx_n}{w_1 + w_2 + w_3 + \cdots + w_n}$$
$x_1, \dots, x_n$ are the values and $w_1, \dots, w_n$ are their weights.
Weight: a number that shows how important, or how large, a value is compared with the others. It can be a real weight (such as kg) or an abstract one (a count, a volume, a time or a ratio).
$x_1$ with weight $w_1$
$x_2$ with weight $w_2$
$x_3$ with weight $w_3$
A value with weight $w$ is counted as if it appeared $w$ times in the data.
- Repeated values: the weighted mean equals the ordinary mean of the longer list in which each value is repeated as many times as its weight.
- Unifies earlier ideas: an average of averages (weights = sizes) and the concentration of a mixture (weights = quantities) are both weighted means.
- Range: the weighted mean lies between the smallest and the largest value.
- Pull of weights: it is pulled toward the value that has the largest weight.
- Equal weights: the weighted mean becomes the ordinary mean.
- Only the ratio of weights matters: multiplying every weight by the same number $k$ gives $\dfrac{k(w_1x_1 + \cdots)}{k(w_1 + \cdots)}$, and $k$ cancels, so the weighted mean is unchanged.
- Adding the same constant to every weight changes the ratio of the weights, so the weighted mean usually changes.
In simple words: a weighted mean is an average in which some values count more than others. The weights tell us by how much.
History of the weighted mean
Brahmagupta (Brāhmasphuṭasiddhānta, c. 628 CE) wrote this formula to define the mean depth of an irregular pit: depth $x_i$ measured over length $w_i$.
Śrīdharācārya (c. 750 CE) used it to find the purity of gold alloys. A purity of $k$ varṇa means $\frac{k}{16}$ of the alloy is gold, and $16$ varṇa $= 24$ karat. Melting pieces together gives a weighted mean of purities, weighted by the units of gold.
The idea reached Europe and became prominent by the 1700s.
4. Custom Weights
Score with weight ratio $w_1 : w_2 : w_3$
$$\text{Score} = \dfrac{w_1 s_1 + w_2 s_2 + w_3 s_3}{w_1 + w_2 + w_3}$$
$s_1, s_2, s_3$ are the scores of the three parts, all on the same scale.
- Why choose weights: an ordinary mean treats every part equally. When some parts should matter more, we assign weights that show their relative importance.
- Weights as a ratio: a ratio such as $3 : 2 : 5$ gives the relative weightage. Equivalent ratios give the same result.
- Same scale first: if the parts are marked out of different totals, convert each to a percentage, $\dfrac{\text{marks obtained}}{\text{total marks}} \times 100$, before applying the weights. Multiplying raw marks by weights gives a wrong answer.
- Rating tables (counts as weights): if a table shows how many customers gave each star rating, the average rating is $\dfrac{\sum (\text{stars} \times \text{number of customers})}{\text{total customers}}$. The number of customers is the weight of each rating.
- Overall rating: combine aspect ratings (for example food, service, ambience) using weights for the aspects. The result lies between the lowest and the highest aspect rating.
- Recent data counts more: to reflect current quality, give newer reviews a larger weight than older ones.
- Comparing without calculating: with the same weights, if one person scores at least as much as another in every part, and more in some part, the weighted total is higher.
- What a scheme captures: a rating system reflects only the aspects included and the importance given to them. Whatever is left out, or weighted too little, is missed.
In simple words: weights show the relative importance of each value with respect to the others. Use a weighted mean whenever the result should reflect how much each part contributes, whether by amount, time or importance.
5. Adding or Removing Values
Mean after adding $k$ new values
$$\text{New mean} = \dfrac{(\text{old mean}) \times n + (\text{sum of new values})}{n + k}$$
Mean after removing $r$ values
$$\text{New mean} = \dfrac{(\text{old mean}) \times n - (\text{sum of removed values})}{n - r}$$
- Idea: treat the existing data as one group (its mean and size $n$) and the new values as another group. The new mean is a weighted mean of the two.
- Average price: for shares, gold or any purchases, average price $= \dfrac{\text{total cost}}{\text{total units}}$. Buying more units is the same as adding values.
- Direction of change: buying at a price below the current average lowers the average; above it raises it; at the same price leaves it unchanged.
- Size matters: the larger the purchase compared with what is already held, the closer the new average moves to the new price. A single value added to a large collection moves the mean only a little.
- Guess before you calculate: the new mean must lie between the old mean and the new value.
- One value changes: if the count stays the same, the sum changes by the same amount as that value, so the mean changes by $\dfrac{\text{change in the value}}{n}$.
- Unknown number of values: if the final mean is given, put the unknown number of added values in the formula, set it equal to the final mean and solve.
- Running averages: a quantity such as a run rate (runs per over) is a total divided by a count, so it updates in the same way each time a new over is added.
6. Cluster-Column Charts
Cluster-column chart: a chart in which bars are grouped into clusters so that bars inside a cluster can be compared side by side.
Clusters = categories
Compare groups
Each cluster is one category and has one bar per group. It is easy to see which group has the most or the least in each category.
Clusters = groups
Compare categories
Each cluster is one group and has one bar per category. It is easy to see which category is biggest or smallest within each group.
- Same data, two charts: a table with two factors can be drawn either way. Choose the clustering by the question that matters most.
- Weakness: a cluster-column chart does not show the total of each cluster or the share of each bar. A stacked bar chart is used when totals also matter.
7. Stacked Bar (or Column) Charts
Stacked bar chart: the bars of a cluster are joined one after another, in a fixed order, to make a single long bar.
The full length is the total; each segment is one component. Only the first segment starts at $0$.
- Full bar: its length stands for the total.
- Each segment: its length stands for one component. Comparing a segment's length with the full bar estimates the fraction or percentage it forms.
- When to use: when you want to compare totals and keep track of the parts that make up each total.
- Reading a segment: only the first segment starts at $0$. For the others, subtract the position where the segment starts from the position where it ends.
- Drawback: since segments do not share a common starting line, comparing segments inside a bar and across bars is harder than in a cluster-column chart. Comparing totals is easier.
- Order: keep the same stacking order in every bar so that colours can be matched easily.
8. 100% Stacked Bar Charts
Share of a part
$$\text{Share (\%)} = \dfrac{\text{Part}}{\text{Total}} \times 100$$
Stretching each bar to the same length keeps the shares but removes all information about the totals.
- Idea: every bar has the same fixed length, standing for $100\%$ of its own total. The segments show each component's share of that total.
- Link with pie charts: a pie chart splits a fixed angle in proportion; a $100\%$ stacked bar splits a fixed length. Lengths in stacked bars are often easier to compare than angles in pies that are turned differently.
- Compares proportions, not absolute values.
- Remaining difficulty: even here, comparison is hard when segments of similar length sit at different positions along their bars.
- Making one from a stacked chart: divide each segment by its bar's total and multiply by $100$.
- The reverse is not possible: from a $100\%$ chart we cannot rebuild the stacked chart, because the totals are lost.
- Both at once: if every bar has the same total (for example $24$ hours in a day), the chart is both a stacked and a $100\%$ stacked bar chart. A single stacked bar can also be read as a $100\%$ bar.
| A 100% stacked bar can tell us | It cannot tell us |
|---|---|
| The largest and smallest part inside one bar The share (fraction or percentage) of each part Which bar has the larger share of a part | The actual numbers Which bar has the larger number of a part Whether two bars have equal or different totals |
9. What Can We Infer from a Chart?
Checking a statement against a 100% stacked bar
1. Parts inside one bar: can be inferred. The total is the same for the whole bar, so a larger share means a larger number.
2. Same part in two different bars: a larger percentage does not mean a larger number, because the other bar's total could be much larger.
3. Claims about totals: cannot be inferred. Many different totals give the same $100\%$ bar.
4. Fair statements across bars: only statements about percentages or shares are safe.
- Same percentages, different counts: two bars with identical shares can come from very different totals, and equal counts can sit under different percentages.
- Why this limits a $100\%$ chart: it helps compare fractions inside each bar but never gives the actual numbers.
- Use both charts when counts matter: a $100\%$ chart shows the mix of responses, a stacked chart shows how many responses there are. A high share of top ratings from very few responses is weak evidence.
| Question | Best chart |
|---|---|
| Which cluster has the larger total? | Stacked bar |
| How big is each part in actual units, and what is the total? | Stacked bar |
| What fraction or percentage is each part of its whole? | 100% stacked bar (or pie chart) |
| Compare composition when totals differ greatly | 100% stacked bar |
| Compare groups within one category by actual value | Cluster column (clusters = categories) |
| Compare categories within one group | Cluster column (clusters = groups) |
10. Reading Data Wisely
- An average can hide differences: the average of a varied collection may describe very few real members. Splitting the data into groups (age, gender, region, income) shows patterns that the single average hides.
- Break data down step by step: a chart by age can be split further by gender; each split answers a more specific question.
- Data depends on time and place: a pattern today may look different decades later, and different countries may show different patterns.
- Before reading any chart, ask: are the bars absolute values or shares? Where does each bar start? Are the totals shown? Are values rounded (so shares may not add to exactly $100\%$)?
- Only infer what the chart supports: if a statement needs actual numbers and the chart shows only percentages, it cannot be concluded.
- Choosing a chart: pick the type that makes the question you care about easiest to answer.
11. All Formulas and Quick Revision
| No. | Concept | Formula |
|---|---|---|
| 1 | Sum from a mean | $\text{Sum} = \text{Mean} \times n$ |
| 2 | Average of two collections | $\dfrac{an + bm}{n + m}$ |
| 3 | Average of equal-sized collections | $\dfrac{a + b + c}{3}$ |
| 4 | Concentration | $\dfrac{\text{Quantity of substance}}{\text{Quantity of mixture}}$ |
| 5 | Weighted mean | $\dfrac{w_1x_1 + \cdots + w_nx_n}{w_1 + \cdots + w_n}$ |
| 6 | Percentage of a component | $\dfrac{\text{marks obtained}}{\text{total marks}} \times 100$ |
| 7 | Mean after adding $k$ values | $\dfrac{(\text{old mean}) \, n + (\text{sum of new values})}{n + k}$ |
| 8 | Share in a 100% stacked bar | $\dfrac{\text{Part}}{\text{Total}} \times 100$ |
Averages
Equal or unequal sizes?
Equal sizes: add the averages and divide by the number of collections. Unequal sizes: use the weighted formula with the sizes as weights.
Charts
Totals or proportions?
Stacked bar: totals and components. 100% stacked bar or pie chart: proportions only. A stacked chart gives a 100% chart, never the reverse.
Check
Estimate first
A weighted mean lies between the smallest and largest value and leans toward the largest weight. Check your answer against this.
Common mistakes:
• Taking the plain average of averages when the collections have different sizes.
• Forgetting to divide by the sum of the weights in a weighted mean.
• Applying weights to marks that are out of different totals without converting to percentages first.
• Using the quantity of one solution instead of the total mixture as the denominator for concentration.
• Concluding that a larger percentage means a larger number when comparing different bars.
• Reading a 100% stacked bar as if it showed totals.
• Reading a middle segment of a stacked bar from $0$ instead of from where it starts.
• Taking the plain average of averages when the collections have different sizes.
• Forgetting to divide by the sum of the weights in a weighted mean.
• Applying weights to marks that are out of different totals without converting to percentages first.
• Using the quantity of one solution instead of the total mixture as the denominator for concentration.
• Concluding that a larger percentage means a larger number when comparing different bars.
• Reading a 100% stacked bar as if it showed totals.
• Reading a middle segment of a stacked bar from $0$ instead of from where it starts.