Data Handling (Mean and Median) | FIO

Question 6

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.

Solution

Let us examine each statement about the mean and median of a set of values.

Step 1 — Understanding Mean and Median

The mean (or average) of a set of values is found by adding all the values together and then dividing by the total count of values.

The median of a set of values is the middle value when the values are arranged in order. If there is an odd number of values, the median is the single middle value. If there is an even number of values, the median is the average of the two middle values.

Step 2 — Analyzing Statement (i)

The statement is: "Removing a value less than the median will decrease the median."

Let us test this with examples.

Case 1: The median stays the same. Let our original set of values be $S = \{1, 2, 3, 3, 4, 5\}$ . First, we arrange them in order, which they already are. There are 6 values (an even number). The two middle values are the 3rd and 4th values, which are 3 and 3. So, the median is the average of these two values: $\text{Median} = \frac{3 + 3}{2}$ $= \frac{6}{2}$

$\boxed{3}$ Now, let us remove a value less than the median (3). We can remove 1. The new set of values is $S' = \{2, 3, 3, 4, 5\}$ . There are 5 values (an odd number). The middle value is the 3rd value, which is 3. So, the new median is 3. In this case, the median stayed the same. It did not decrease.

Case 2: The median increases. Let our original set of values be $S = \{1, 2, 3, 4, 5\}$ . The median is the 3rd value, which is 3. Let us remove a value less than the median (3). We can remove 1. The new set of values is $S' = \{2, 3, 4, 5\}$ . There are 4 values (an even number). The two middle values are the 2nd and 3rd values, which are 3 and 4. So, the new median is the average of these two values: $\text{New Median} = \frac{3 + 4}{2}$ $= \frac{7}{2}$

$\boxed{3.5}$ In this case, the median increased from 3 to 3.5. It did not decrease.

So, the statement is not always true. It can sometimes stay the same or increase. It can also decrease in some situations, but we have shown it is not always true.

Therefore, the statement is Sometimes true.

Step 3 — Analyzing Statement (ii)

The statement is: "Including a value less than the mean will decrease the mean."

Let us say we have $N$ values, and their sum is $S$ . The original mean is $M$ . $M = \frac{S}{N}$ Now, we include a new value, let us call it $x$ . We are told that $x$ is less than the original mean, so $x < M$ . The new sum of values will be $S' = S + x$ . The new number of values will be $N' = N + 1$ . The new mean will be $M' = \frac{S+x}{N+1}$ .

We want to check if $M' < M$ . Let us substitute the expressions: $\frac{S+x}{N+1} < \frac{S}{N}$ We can multiply both sides by $N(N+1)$ (which is a positive number, so the inequality direction does not change): $N(S+x) < S(N+1)$ Let us expand both sides: $NS + Nx < SN + S$ Now, we can subtract $NS$ from both sides: $Nx < S$ Finally, we can divide both sides by $N$ (which is a positive number): $x < \frac{S}{N}$ We know that $S/N$ is the original mean $M$ . So, the condition for the new mean to be less than the original mean is $x < M$ . This is exactly what the statement says: "Including a value less than the mean". Since this mathematical relationship always holds true, the statement is Always true.

Step 4 — Analyzing Statement (iii)

The statement is: "Including any 4 values will not affect the median."

Let us test this with examples.

Case 1: The median changes. Let our original set of values be $S = \{1, 2, 3\}$ . The median is 3. Now, we include 4 new values: $\{10, 11, 12, 13\}$ . The new set of values, sorted, is $S' = \{1, 2, 3, 10, 11, 12, 13\}$ . There are $3+4=7$ values. The median is the middle value, which is the 4th value. The new median is 10. In this case, the median changed from 3 to 10.

Case 2: The median stays the same. Let our original set of values be $S = \{1, 2, 3, 4, 5\}$ . The median is 3. Now, we include 4 new values: $\{3, 3, 3, 3\}$ . The new set of values, sorted, is $S' = \{1, 2, 3, 3, 3, 3, 3, 4, 5\}$ . There are $5+4=9$ values. The median is the middle value, which is the 5th value. The new median is 3. In this case, the median stayed the same.

Since the median can either change or stay the same, the statement "will not affect the median" is not always true and not never true.

Therefore, the statement is Sometimes true.

Step 5 — Analyzing Statement (iv)

The statement is: "Including 4 values less than the median will increase the median."

Let us test this with examples.

Case 1: The median decreases. Let our original set of values be $S = \{10, 20, 30, 40, 50\}$ . The median is the 3rd value, which is 30. Now, we include 4 values that are all less than the median (30). Let us choose $\{1, 2, 3, 4\}$ . The new set of values, sorted, is $S' = \{1, 2, 3, 4, 10, 20, 30, 40, 50\}$ . There are $5+4=9$ values. The median is the middle value, which is the 5th value. The new median is 10. In this case, the median decreased from 30 to 10.

Case 2: Another example where the median decreases. Let our original set of values be $S = \{10, 20, 30, 40\}$ . The median is the average of the two middle values (20 and 30). $\text{Median} = \frac{20 + 30}{2}$ $= \frac{50}{2}$

$\boxed{25}$ Now, we include 4 values that are all less than the median (25). Let us choose $\{1, 2, 3, 4\}$ . The new set of values, sorted, is $S' = \{1, 2, 3, 4, 10, 20, 30, 40\}$ . There are $4+4=8$ values. The two middle values are the 4th and 5th values, which are 4 and 10. The new median is the average of these two values: $\text{New Median} = \frac{4 + 10}{2}$ $= \frac{14}{2}$ $\boxed{7}$ In this case, the median decreased from 25 to 7.

When we add values that are smaller than the current median, these new values will take up positions at the lower end of the sorted list. This pushes the original values (including the median) further down the list. This usually results in the median decreasing, or at best, staying the same if the original median was a repeated value and the new values don't shift its position. It will never increase.

Therefore, the statement is Never true.

Answer

(i) Sometimes true: Removing a value less than the median can decrease the median. But if the data set has an even number of elements and the removed value is below the lower of the two middle values, the median may stay the same. (ii) Always true: The mean is the sum of all values divided by the number of values. Adding a value less than the mean decreases the total sum less than proportionally to the increase in the number of values, thus decreasing the mean. (iii) Sometimes true: Including any 4 values could shift the median depending on whether those values are above or below the median and on the original number of observations. (iv) Never true: Including values less than the median will either keep the median the same or decrease it, but it will never increase it.

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?