Class 9 Maths Ganita Manjari Chapter 10 End of Chapter Exercises How Quantities Combine: Understanding Data

Solution:

The graph is a 100% stacked bar chart showing how different types of disability are distributed within four age groups in India, according to Census 2011. Since every bar represents 100%, the graph compares proportions, not the actual number of persons.

Some observations and inferences are:

• Seeing-related disability forms a noticeably larger share in the 60 years and above age group than in the younger age groups.
• Movement-related disability also becomes more prominent in the older age groups.
• Hearing-related disability forms a fairly large share across all age groups, though its proportion is lower among those aged 60 and above than in some younger groups.
• Speech-related disability accounts for a relatively small share in every age group.
• The category labelled “mental retardation” in the Census 2011 graph (now generally referred to as intellectual disability) forms a larger share among younger age groups and a smaller share among older age groups.
• Multiple disability has a comparatively high share in the 60 years and above group.
• The category “any other” accounts for a substantial share among the younger and middle-age groups but is much smaller for those aged 60 and above.

What we may wonder

We may wonder why seeing, movement, and multiple disabilities form larger shares among older people, whether this is related to ageing, and whether the pattern has changed since the 2011 Census.

We may also want to know the actual number of persons in each age group. This graph alone cannot tell us that because it shows only percentages.

(i) Recall the previous day and fill in the approximate time spent lying down, sitting, and standing/moving around, etc. Ask at least 2 of your family members about their day and fill it in. Alternatively, you can choose to track just your own body-state for any one of the weekdays, Saturday, and Sunday. Visualise it using a stacked bar chart. Discuss with your class how you arrived at the estimated time spent in each state.
(ii) Visualise your family’s monthly expenditure using a 100% stacked bar chart.
(a) First, identify the expense categories in your family budget.
(b) Discuss with your family members to decide how you will collect and organise the data.
(c) Collect expense data for at least 3 months.
(d) Represent the data in a 100% stacked bar chart, with each bar showing how the total monthly expenditure is divided across the different categories.
(e) Finally, write your observations and inferences based on the chart.
Answer:

Solution:

Since this is an individual project, the actual data will differ from person to person. The following can be used as a sample project.

(i) Time spent in different body-states

I estimated the time spent by myself and two family members during a typical day.

Person	Lying down	Sitting	Standing/Moving	Total
Myself	8 hours	9 hours	7 hours	24 hours
Mother	7.5 hours	8.5 hours	8 hours	24 hours
Father	7 hours	10 hours	7 hours	24 hours

A stacked bar chart can be drawn with one bar for each person. Each bar should extend from 0 to 24 hours and be divided into the three body-states.

I estimated the times by recalling sleeping hours, time spent studying or working while sitting, meal and travel time, and time spent walking, exercising or doing household activities.

Observations: My father spent the most time sitting, while my mother spent the most time standing or moving around. All three spent around 7–8 hours lying down. The stacked bar chart makes it easy to compare how each person's 24-hour day is divided.

Since this is an individual project, the actual expenditure will vary from family to family. The following is a sample project showing how it can be completed.

(a) Identify the expense categories

The monthly family expenditure can be divided into categories such as house rent, food and groceries, utilities, education, transport, and other expenses.

(b) Decide how to collect and organise the data

The expenditure data can be collected by discussing monthly spending with family members and referring to bills, receipts, bank statements, and digital payment records. The expenses can then be grouped under the selected categories and recorded month-wise in a table.

(c) Collect expense data for at least 3 months

Suppose the following expenditure was recorded:

Expense category	January	February	March
House rent	₹15,000	₹15,000	₹15,000
Food & groceries	₹12,000	₹14,000	₹13,000
Utilities	₹5,000	₹6,000	₹5,500
Education	₹8,000	₹10,000	₹9,000
Transport	₹5,000	₹6,000	₹5,500
Other expenses	₹5,000	₹9,000	₹7,000
Total expenditure	₹50,000	₹60,000	₹55,000

(d) Represent the data using a 100% stacked bar chart

To draw the 100% stacked bar chart, first convert every expense into a percentage of that month's total expenditure:

Percentage expenditure = (Amount spent on category / Total monthly expenditure) × 100

Expense category	January	February	March
House rent	30%	25%	27.27%
Food & groceries	24%	23.33%	23.64%
Utilities	10%	10%	10%
Education	16%	16.67%	16.36%
Transport	10%	10%	10%
Other expenses	10%	15%	12.73%
Total	100%	100%	100%

Family monthly expenditure

100% stacked bar chart showing the share of each expense category over three months.

(e) Each bar represents 100% of the expenditure for one month, and the different sections show the proportion spent on each category.

(i) Discuss and arrive at 4–6 aspects to rate. For each aspect chosen, justify why it matters for an overall experience.
(ii) Discuss and decide the relative weights and justify.
(iii) Collect or imagine ratings from at least 10 people on a scale of 1–5 (5 being very good).
(iv) Compute each individual rating. Compute the overall average across individuals.
(v) Make a 100% stacked bar chart using your data.
(vi) Write a short note on the observations and inferences, what your rating system captures well and what it misses.

Answer:

Small-Group Project: Clinic/Hospital Experience

For this sample project, we chose (d) Clinic/Hospital Experience.

(i) Aspects to rate and why they matter

We selected five aspects:

Aspect	Why it matters
Doctor consultation	The quality of diagnosis, explanation and treatment is the most important part of the visit.
Staff behaviour	Polite and helpful staff improve the patient's overall experience.
Waiting time	Long waiting times can make the visit inconvenient and stressful.
Cleanliness	Hygiene is particularly important in a healthcare facility.
Facilities	Availability and quality of seating, washrooms, equipment and other facilities affect comfort and convenience.

(ii) Relative weights

After discussion, we assigned the following weights:

Aspect	Weight
Doctor consultation	30%
Staff behaviour	20%
Waiting time	20%
Cleanliness	15%
Facilities	15%
Total	100%

We gave the highest weight to doctor consultation because medical care is the main purpose of visiting a clinic or hospital. Staff behaviour and waiting time were given the next-highest weights, while cleanliness and facilities were also considered important.

(iii) Ratings collected from 10 people

Ratings are on a scale of 1 to 5, where 5 means very good.

Person	Doctor	Staff	Waiting time	Cleanliness	Facilities
1	5	4	3	5	4
2	4	4	4	4	4
3	5	5	4	4	5
4	3	4	2	4	3
5	4	3	3	5	4
6	5	4	4	5	5
7	4	5	3	4	4
8	3	3	2	3	3
9	4	4	5	4	4
10	5	4	3	4	5

(iv) Individual ratings and overall average

The weighted rating is calculated as:

Individual rating = (30 × Doctor + 20 × Staff + 20 × Waiting + 15 × Cleanliness + 15 × Facilities) / 100

For Person 1:

= (30 × 5 + 20 × 4 + 20 × 3 + 15 × 5 + 15 × 4) / 100
= 425 / 100
= 4.25

Similarly:

Person	Final weighted rating
1	4.25
2	4.00
3	4.65
4	3.15
5	3.75
6	4.60
7	4.00
8	2.80
9	4.20
10	4.25

Overall average:

= (4.25 + 4.00 + 4.65 + 3.15 + 3.75 + 4.60 + 4.00 + 2.80 + 4.20 + 4.25) / 10

= 39.65 / 10

= 3.965 ≈ 3.97 out of 5

Therefore, the overall rating of the clinic/hospital is approximately 3.97 out of 5.

(v) 100% stacked bar chart

For the chart, we can show the percentage of respondents who gave 1-star, 2-star, 3-star, 4-star and 5-star ratings to each aspect.

Aspect	1★	2★	3★	4★	5★
Doctor consultation	0%	0%	20%	40%	40%
Staff behaviour	0%	0%	20%	60%	20%
Waiting time	0%	20%	40%	30%	10%
Cleanliness	0%	0%	10%	60%	30%
Facilities	0%	0%	20%	50%	30%

(vi) Observations and inferences

The overall rating of about 3.97 out of 5 indicates that the respondents generally had a positive experience. Doctor consultation received particularly strong ratings, with 80% of respondents giving it either 4 or 5. Cleanliness and facilities were also rated positively.

Waiting time appears to be the main area for improvement, as it received the largest share of 2- and 3-star ratings.

The rating system captures important factors such as medical care, staff behaviour, waiting time, cleanliness and facilities. However, it does not capture everything—for example, treatment cost, availability of medicines, emergency services, accessibility, or individual medical outcomes. The results may also change if a larger group of people is surveyed.

14. Whole class project: Each student shares the average age of their family and the number of family members. Discuss among the class and come up with a way to find out the average age of all the families of the class.

Answer:

To find the average age of all family members represented in the class, use the number of members in each family as the weight.

• Record each family’s average age and number of members.
• Multiply the average age by the number of members to find the total age of that family.
• Add the total ages of all families.
• Divide by the total number of family members.

If the family averages are a₁, a₂, …, aₙ and the family sizes are n₁, n₂, …, nₙ, then:

Combined average age = (n₁a₁ + n₂a₂ + … + nₙaₙ) / (n₁ + n₂ + … + nₙ)

For example:

• Family A: 4 members, average age = 25 years
  Total age = 4 × 25 = 100 years
• Family B: 6 members, average age = 30 years
  Total age = 6 × 30 = 180 years
• Family C: 5 members, average age = 28 years
  Total age = 5 × 28 = 140 years

Therefore,

Combined average = (100 + 180 + 140) / (4 + 6 + 5)

= 420 / 15

= 28 years

Therefore, the average age of all the family members is 28 years.

15. Given some data with corresponding weights, what would happen to the weighted average if all the weights are increased by a constant value, say 1? If required, experiment with some data. What do you observe? Justify your answer using algebra.
Answer:

Unlike multiplying all the weights by the same number, adding 1 to every weight will generally change the weighted average.

For example, take the data values 10 and 20 with weights 1 and 3.

Original weighted average:

= (10 × 1 + 20 × 3) / (1 + 3)
= 70 / 4
= 17.5

Now increase each weight by 1. The new weights are 2 and 4.

New weighted average:

= (10 × 2 + 20 × 4) / (2 + 4)
= 100 / 6
≈ 16.67

So, the weighted average has changed.

Algebraic justification

Suppose the data values are x₁, x₂, …, xₙ with weights w₁, w₂, …, wₙ.

Original weighted average:

A = (w₁x₁ + w₂x₂ + … + wₙxₙ) / (w₁ + w₂ + … + wₙ)

If 1 is added to every weight, the new weighted average becomes:

A' = [(w₁ + 1)x₁ + (w₂ + 1)x₂ + … + (wₙ + 1)xₙ] / [(w₁ + 1) + (w₂ + 1) + … + (wₙ + 1)]

Simplifying:

A' = (w₁x₁ + w₂x₂ + … + wₙxₙ + x₁ + x₂ + … + xₙ) / (w₁ + w₂ + … + wₙ + n)

This is generally different from the original weighted average.

Observation: Adding the same positive number to all the weights makes the weights more similar to one another. Therefore, the new weighted average moves closer to the ordinary arithmetic mean of the data.

Hence, increasing all weights by 1 generally changes the weighted average; it does not remain unchanged.