Data Handling (Mean and Median) | IT

Question 6

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?

Solution

The special 'centre' described in the question is actually the arithmetic mean of all the values in the dataset.

Step 1 — Understanding the 'centre' condition

Let us represent the given values as $x_1, x_2, \dots, x_n$ . Let $C$ be the 'centre' we are looking for. The distance between $C$ and a value $x_i$ is $|C - x_i|$ . The question states a special condition for $C$ . It says the sum of distances to values lower than $C$ must equal the sum of distances to values higher than $C$ . Let us write this condition using mathematical symbols. If a value $x_i$ is lower than $C$ , its distance from $C$ is $C - x_i$ . If a value $x_j$ is higher than $C$ , its distance from $C$ is $x_j - C$ . So, the condition for $C$ to be a 'centre' is: $\sum_{x_i < C} (C - x_i) = \sum_{x_j > C} (x_j - C)$ This equation defines our 'centre'.

Diagram 1

Step 2 — Simplifying the condition

Let us expand the sums on both sides of the equation. The left side is the sum of $C$ for each value less than $C$ , minus the sum of those values. The right side is the sum of values greater than $C$ , minus the sum of $C$ for each of those values. Let $N_L$ be the count of values that are smaller than $C$ . Let $N_H$ be the count of values that are larger than $C$ . So, our equation from Step 1 becomes: $N_L \cdot C - \sum_{x_i < C} x_i = \sum_{x_j > C} x_j - N_H \cdot C$ Now, let us rearrange the terms to solve for $C$ . We move all terms that contain $C$ to one side of the equation. We move all other terms to the other side. $N_L \cdot C + N_H \cdot C = \sum_{x_i < C} x_i + \sum_{x_j > C} x_j$ We can factor out $C$ from the left side. $(N_L + N_H) C = \sum_{x_i < C} x_i + \sum_{x_j > C} x_j$ Let $N_E$ be the count of values that are exactly equal to $C$ . The total number of values in our dataset is $n$ . So, $n = N_L + N_H + N_E$ . The sum of all values in the dataset is $\sum_{k=1}^n x_k$ . This total sum can be split into three parts: $\sum_{k=1}^n x_k = \sum_{x_i < C} x_i + \sum_{x_k = C} x_k + \sum_{x_j > C} x_j$ The sum of values equal to $C$ is simply $N_E \cdot C$ . So, we can write the total sum as: $\sum_{k=1}^n x_k = \left( \sum_{x_i < C} x_i + \sum_{x_j > C} x_j \right) + N_E \cdot C$ From our simplified condition, we know that the bracketed term is equal to $(N_L + N_H) C$ . Let us substitute this back into the equation for the total sum. $\sum_{k=1}^n x_k = (N_L + N_H) C + N_E \cdot C$ We can factor out $C$ again from the right side. $\sum_{k=1}^n x_k = (N_L + N_H + N_E) C$ Since $N_L + N_H + N_E$ is the total number of values, $n$ : $\sum_{k=1}^n x_k = n \cdot C$ Finally, we can divide both sides by $n$ to find the value of $C$ . $C = \frac{\sum_{k=1}^n x_k}{n}$

$\boxed{C = \text{Arithmetic Mean}}$ This shows that the 'centre' described in the question is always the arithmetic mean (or average) of all the values.

Step 3 — Concluding uniqueness

The arithmetic mean is a specific calculation. For any given set of numbers, there is only one unique value for its arithmetic mean. For example, the mean of the numbers $\{2, 4, 6\}$ is $4$ . It cannot be any other number. Since the 'centre' must be equal to the arithmetic mean, there can only be one such 'centre' for any given set of values.

Answer

(i) No, there can be only one such 'centre'. (ii) This 'centre' is the arithmetic mean of all the values in the dataset. (iii) The arithmetic mean is a unique value for any given set of numbers, so there can only be one such 'centre'.

More questions in IT

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

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

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

Mark the mean for the collections below.

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

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.

Explore if it is possible to include or remove 2 values such that the mean is unchanged.

You may use the following data to experiment with.

Q10

How about including or removing 3 values without changing the mean? Is it possible?

Q12

Try to include 2 values greater than the mean and 1 value less than the mean, so that the mean stays the same.

Q13

We saw what happens to the mean when values are included or removed from the collection. What happens to the mean if every value in the collection increases by some fixed number?

Q14

Consider the data: 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5. Calculate its mean.

Q15

Now, consider this data with every value increased by 10: 18, 13, 20, 23, 14, 16, 17, 17, 18, 18, 15. What is its mean? Is there a quicker way to find out?

[Hint: Observe the following dot plots corresponding to the two data collections.]

Q16

Try to explain, using algebra, what the average is when a fixed number, e.g., 2 is subtracted from every value in the collection.

Q17

Try to explain this using the fair-share interpretation of average that you learnt last year.

Q18

What happens to the average if every value in the collection is doubled?

Q20

Can you tell what data is in column B7?

Q21

In which subjects has Ashwin scored more than 30 marks?

Q22

Context: Sudhakar has collected the mid-term exam marks obtained by his Grade 8 students in a spreadsheet.

Q. What formula would you type to find out the class average marks in Science?

Q23

Context: Sudhakar has collected the mid-term exam marks obtained by his Grade 8 students in a spreadsheet.

Q. Find out if the class average marks in Odia is greater than the class average marks in Telugu.

Q24

Context: Sudhakar has collected the mid-term exam marks obtained by his Grade 8 students in a spreadsheet.

Q. Show the average marks in other subjects after the last row by typing the appropriate formulae.

Q25

Context: Sudhakar has collected the mid-term exam marks obtained by his Grade 8 students in a spreadsheet.

Q. Get the total scores of each student by typing the appropriate formulae.

Q26

Context: The following figure is a clustered-column graph that shows the monthly maximum temperature in Kerala and Punjab in 2023.

Q. Now, observe the following graph. Do both these graphs represent the same information?

Q27

How do we get the maximum temperature over a month in a state?

Q28

Notice how the graph is organised, what scale is used, and what patterns the data shows.

Q29

Analyse and interpret each of the observations you made. Share appropriate summary/conclusion statements.

Q30

Here are some directions to think about —

Why are the trends so distinct in these two states? What factors determine a region's temperature?
You might be curious to look at the trends of a few other states. Are there other types of trends that states exhibit?
Which states show trends similar to Punjab's? Is there anything common between these states?

Q31

Context: After analyzing the monthly maximum temperature trends for Kerala and Punjab,

Q. What thoughts or questions occur to you?

Q32

Context: Space Jam: A Traffic Problem in the Future?

Take a look at the line graph below showing the annual number of objects launched into space.

Q. What could be the possible method used to derive this data? Discuss.

Q33

Context: Space Jam: A Traffic Problem in the Future?

Step 1: Identify what is given

Q. Notice how the graph is organised, what scale is used, and what patterns the data shows.

Q34

Context: Space Jam: A Traffic Problem in the Future?

Step 2: Infer from and interpret what is given

Q. Analyse and interpret each of the observations you made. Once all interpretations are made, summarising/concluding statements can be made.

Q35

Which of the following statements are valid inferences?

From 2012 till 2024, the worldwide count of space object launches increased every year.
USA is a major contributor in the years 2022–24, launching about $\frac{3}{4}$ th of the worldwide count.
Nepal did not launch any object in the period 2012–24.
The combined count of object launches by China and Russia in 2024 is about 400.

Q36

Identify two consecutive years where the worldwide count increased by 2 times or more.

Q37

What could be the possible method to compile this data?

Q38

Mark these cities on a map of India. What is common to how they are grouped in the graphs? Share your observations and inferences about the graphs.

Q39

Identify the peak months and low months of rainfall for each city.

Q40

Read about the south-west monsoon and north-east monsoon and which regions come under the influence of these and when.

Q41

Share your observations. Based on this infographic, answer the following:

(i) The value of Karnataka is hidden. Can you guess what it could be?

(ii) Which are the top 5 states where rice is the most popular?

(iii) Which are the top 5 states where wheat is the most popular?

(iv) List a few states where the preference between rice and wheat is more or less balanced.

Q42

Context: Manoj records his activities throughout the day by colouring a strip of paper with 48 boxes, marking time in 30 minute intervals from midnight to midnight. He has recorded five types of activities for three different days of the week on three coloured strips: (i) Sleeping (ii) Eating (iii) Meeting friends, hobbies, media, time with family (iv) Attending classes, studying and homework (v) Showering and getting dressed, yoga or exercise (vi) Travelling

Q. Look at the three coloured strips carefully.

(i) What activity does each colour stand for?

(ii) The three strips correspond to the days Friday – Sunday in some order. Which day do you think each strip represents?

(iii) On one of these days, he went out with friends to watch a long movie. When do you think this happened?

(iv) At what time does his school break for lunch?

(v) What more can the strips tell us?

Q43

What would your strip for a weekday look like? How similar or different is it to Manoj’s?

Q44

What would a strip of your typical day during your vacation look like? How similar/different would it look?

Q45

What would a strip for any of the adults in your family look like? Make a strip of a day for any adult at home. Compare your strip with theirs. What do you find interesting?

Q46

Share your observations on this graph. What do you find interesting?

Q47

Referring to the graph below, which of the following statements are valid? Why?

(i) In 1983, the majority in rural areas used kerosene as a primary lighting source while the majority in urban areas used electricity.

(ii) The use of kerosene as a primary lighting source has decreased over time in both rural and urban areas.

(iii) In the year 2000, 10% of the urban households used electricity as a primary lighting source.

(iv) In 2023, there were no power cuts.

Q48

Answer the following questions based on the line graph.

(i) How long do children aged 10 in urban areas spend each day on hobbies and games?

(ii) At what age is the average time spent daily on hobbies and games by rural kids 1.5 hours?

(a) 8 years

(b) 10 years

(d) 14 years

(e) 18 years

(iii) Are the following statements correct?

(a) The average time spent daily on hobbies and games by kids aged 15 is twice that of kids aged 10.

(b) All rural kids aged 15 spend at least 1 hour on hobbies and games everyday.

Q49

We all know the typical sunrise and sunset timings. Do you know when the moon rises and sets? Does it follow a regular pattern like the sun? Let's find out. The following graph shows the moonrise and moonset time over a month:

(i) Find out on what dates amavasya (new moon) and purnima (full moon) were in this month.

(ii) What do you notice? What do you wonder?