Data Handling (Mean and Median) | FIO

Question 15

Mean Grids:

(i) Fill the grid with 9 distinct numbers such that the average along each row, column, and diagonal is 10.

(ii) Can we fill the grid by changing a few numbers and still get 10 as the average in all directions?

Solution

For the average of three numbers to be 10, their sum must be 30.

Step 1 — Calculate the required sum

The problem asks for the average of numbers in each row, column, and diagonal to be 10. Each row, column, and diagonal has 3 numbers. To find the sum, we multiply the average by the number of elements. $\text{Sum} = \text{Average} \times \text{Number of elements}$ $= 10 \times 3$

$\boxed{30}$ So, every row, column, and diagonal in the grid must add up to 30.

Diagram 1

Step 2 — Fill the grid (Part i)

We need to fill the 3x3 grid with 9 distinct numbers. The sum of numbers in each row, column, and diagonal must be 30. This type of grid is called a magic square. A common 3x3 magic square uses numbers from 1 to 9. In that standard magic square, the sum of each row, column, and diagonal is 15. The standard magic square is: $\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ We want the sum to be 30, which is twice the standard sum of 15. So, we can multiply each number in the standard magic square by 2. This will give us 9 distinct numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18. Let's multiply each number by 2: $\begin{pmatrix} 8 \times 2 & 1 \times 2 & 6 \times 2 \\ 3 \times 2 & 5 \times 2 & 7 \times 2 \\ 4 \times 2 & 9 \times 2 & 2 \times 2 \end{pmatrix}$ $= \begin{pmatrix} 16 & 2 & 12 \\ 6 & 10 & 14 \\ 8 & 18 & 4 \end{pmatrix}$ Let's check the sums for this grid: Row 1: $16 + 2 + 12 = \mathbf{30}$ Row 2: $6 + 10 + 14 = \mathbf{30}$ Row 3: $8 + 18 + 4 = \mathbf{30}$ Column 1: $16 + 6 + 8 = \mathbf{30}$ Column 2: $2 + 10 + 18 = \mathbf{30}$ Column 3: $12 + 14 + 4 = \mathbf{30}$ Diagonal 1 (top-left to bottom-right): $16 + 10 + 4 = \mathbf{30}$ Diagonal 2 (top-right to bottom-left): $12 + 10 + 8 = \mathbf{30}$ All sums are 30, and all 9 numbers are distinct.

Step 3 — Changing numbers (Part ii)

Yes, we can fill the grid by changing some numbers and still get an average of 10 in all directions. The arrangement of numbers in a magic square is not unique. If we change a number in one cell, we can adjust other numbers in its row, column, and diagonals. This adjustment ensures all sums remain 30. For example, we can rotate the entire grid from Step 2. Let's rotate the grid 90 degrees clockwise: Original Grid: $\begin{pmatrix} 16 & 2 & 12 \\ 6 & 10 & 14 \\ 8 & 18 & 4 \end{pmatrix}$ Rotated Grid: $\begin{pmatrix} 8 & 6 & 16 \\ 18 & 10 & 2 \\ 4 & 14 & 12 \end{pmatrix}$ This new grid has different numbers in most positions. However, all its row, column, and diagonal sums are still 30. This means the average in all directions is still 10. So, there are multiple ways to fill the grid.

Answer

(i) To get an average of 10, the sum of each row, column, and diagonal must be 30. One possible grid is: Row 1: 16, 2, 12 Row 2: 6, 10, 14 Row 3: 8, 18, 4 (ii) Yes, we can fill the grid by changing a few numbers and still get 10 as the average in all directions. This is because there are multiple ways to arrange numbers to form such a grid. If a number is changed, other numbers can be adjusted to maintain the sums of 30.

More questions in FIO

Find the mean of the following data and share your observations:

(i) The first 50 natural numbers. (ii) The first 50 odd numbers. (iii) The first 50 multiples of 4.

The dot plot below shows a collection of data and its average; but one dot is missing. Mark the missing value so that the mean is 9 (as shown below).

Sudhakar, the class teacher, asks Shreyas to measure the heights of all 24 students in his class and calculate the average height. Shreyas informs the teacher that the average height is 150.2 cm. Sudhakar discovers that the students were wearing uniform shoes when the measurements were taken and the shoes add 1 cm to the height.

(i) Should the teacher get all the heights measured again without the shoes to find the correct average height? Or is there a simpler way?

(ii) What is the correct average height of the class?

(a) 174.2 cm

(b) 126.2 cm

(d) 149.2 cm

(e) 151.2 cm

(f) None of the above

(g) Insufficient information

The three dot plots below show the lengths, in minutes, of songs of different albums. Which of these has a mean of 5.57 minutes? Explain how you arrived at the answer.

Find the median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92.

(i) If we include one value to the data (in the given list) without affecting the median, what could that value be?

(ii) If we include two values to the data without affecting the median what could the two values be?

(iii) If we remove one value from the data without affecting the median what could the value be?

Examine the statements below and justify if the statement is always true, sometimes true, or never true.

(i) Removing a value less than the median will decrease the median.

(ii) Including a value less than the mean will decrease the mean.

(iii) Including any 4 values will not affect the median.

(iv) Including 4 values less than the median will increase the median.

The mean of the numbers 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find the value of y.

The mean of a set of data with 15 values is 134. Find the sum of the data.

Consider the data: 12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p. Which of the following number(s) could be p if the median of this data is 29?

(i) 10 (ii) 25 (iii) 40 (iv) 100 (v) 29 (vi) 47 (vii) 30

Q10

The number of times students rode their cycles in a week is shown in the dot plot below. Four students rode their cycles twice in that week.

(i) Find the average number of times students rode their cycles. (ii) Find the median number of times students rode their cycles. (iii) Which of the following statements are valid? Why? (a) Everyone used their cycle at least once. (b) Almost everyone used their cycle a few times. (c) There are some students who cycled more than once on some days. (d) Exactly 5 students have used their cycles more than once on some days. (e) The following week, if all of them cycled 1 more time than they did the previous week, what would be the average and median of the next week's data?

Q11

A dart-throwing competition was organised in a school. The number of throws participants took to hit the bull's eye (the centre circle) is given in the table below. Describe the data using its minimum, maximum, mean and median.

Q12

The average number of customers visiting a shop and the average number of customers actually purchasing items over different days of the week is shown in the table below. Visualise this data on a line graph.

Q13

The average number of days of rainfall in each month for a few cities is shown in the table below:

(i) What could be the possible method to compile this data?

(ii) Mark the data for Mangaluru, Port Blair, and Rameswaram in the line graph shown below. You can round off the values to the nearest integer.

(iii) Based on the line for New Delhi in the graph fill the data in the table.

(iv) Which city among these receives the most number of days of rainfall per year? Which city gets the least number of days of rainfall per year?

(v) Looking at the table, when is the rainy season in New Delhi and Rameswaram?

Q14

The following line graph shows the number of births in every month in India over a time period:

(i) What are your observations?

(ii) What was the approximate number of births in July 2017?

(iii) What time period does the graph capture?

(iv) Compare the number of births in the month of January in the years 2018, 2019, and 2020.

(v) Estimate the number of births in the year 2019.

Q15

Mean Grids:

(i) Fill the grid with 9 distinct numbers such that the average along each row, column, and diagonal is 10.

(ii) Can we fill the grid by changing a few numbers and still get 10 as the average in all directions?

Q16

Give two examples of data that satisfy each of the following conditions:

(i) 3 numbers whose mean is 8.

(ii) 4 numbers whose median is 15.5.

(iii) 5 numbers whose mean is 13.6.

(iv) 6 numbers whose mean = median.

(v) 6 numbers whose mean > median.

Q17

Fill in the blanks such that the median of the collection is $13$ : $5, 21, 14, \underline{\quad}, \underline{\quad}, \underline{\quad}$ . How many possibilities exist if only counting numbers are allowed?

Q18

Fill in the blanks such that the mean of the collection is $6.5$ : $3$ , $11$ , $\underline{\quad}$ , $\underline{\quad}$ , $15$ , $6$ . How many possibilities exist if only counting numbers are allowed?

Q19

Check whether each of the statements below is true. Justify your reasoning. Use algebra, if necessary, to justify.

(i) The average of two even numbers is even.

(ii) The average of any two multiples of 5 will be a multiple of 5.

(iii) The average of any 5 multiples of 5 will also be a multiple of 5.

Q20

There were 2 new admissions to Sudhakar’s class just a couple of days after the class average height was found to be 150.2 cm.

(i) Which of the following statements are correct? Why? (a) The average height of the class will increase as there are 2 new values. (b) The average height of the class will remain the same. (c) The heights of the new students have to be measured to find out the new average height. (d) The heights of everyone in the class has to be measured again to calculate the new average height.

(ii) The heights of the two new joinees are 149 cm and 152 cm. Which of the following statements about the class’ average height are correct? Why? (a) The average will remain the same. (b) The average will increase. (c) The average will decrease. (d) The information is not sufficient to make a claim about the average.

(iii) Which of the following statements about the new class average height are correct? Why? (a) The median will remain the same. (b) The median will increase. (c) The median will decrease. (d) The information is not sufficient to make a claim about median.

Q21

Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.

Q22

The weights of people in a group were measured every month. The average weight for the previous month was 65.3 kg and the median weight was 67 kg. The data for this month showed that one person has lost 2 kg and two have gained 1 kg. What can we say about the change in mean weight and median weight this month?

Q23

The following table shows the retail price (in ₹) of iodised salt in the month of January in a few states over 10 years. For your calculations and plotting you may round off values to the nearest counting number.

(i) Choose data from any 3 states you find interesting and present it through a line graph using an appropriate scale.

(ii) What do you find interesting in this data? Share your observations.

(iii) Compare the price variation in Gujarat and Uttar Pradesh.

(iv) In which state has the price increased the most from 2016 to 2025?

(v) What are you curious to explore further?

Q24

The following graphs show the sunrise and sunset times across the year at 4 locations in India. Observe how the graphs are organised. Are you able to identify which lines indicate the sunrise and which indicate the sunset?

Answer the following questions based on the graphs:

(i) At which place does the sun rise the earliest in January? What is the approximate day length at this place in January?

(ii) Which place has the longest day length over the year?

(iii) Share your observations—what do you find interesting? What are you curious to find out?