Data Handling (Mean and Median)

Individual project: Make your own activity strip for different days of the week.

(i) Do you eat and sleep at regular times every day? Typically how long do you spend outdoors?

(ii) Calculate the average time spent per activity. Represent this average day using a strip.

(iii) Similarly, track the activities of any adult at home. Compare your data with theirs.

Small group project: Make a group of 3–4 members. Do at least one of the following:

(i) Track daily sleep time of all your family members for a week. Daily sleep time includes night sleep, naps, and any sleep during the day.

(a) Represent this on strips.

(b) Put together the data of all your group members. Calculate the average and median sleep time of children, adults, elderly.

(c) Share your findings and observations.

(ii) When do schools start and end? On a weekday, Manoj’s school starts at 9:30 am and ends at 4:30 pm, i.e., 7 hours which include class time and breaks. Collect information on the daily timings of different schools for Grade 8, including class time and break time (the schools can be anywhere in the country. You can ask your neighbours, relatives, parents and friends to find out). Analyse and present the data collected.

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

(c) 150.2 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

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?

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.

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.

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?

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.

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?

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.

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?

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?

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.

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.

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

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?

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?

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?

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.

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?

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.

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

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

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?

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

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.]

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

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

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

Can you tell what data is in column B7?

In which subjects has Ashwin scored more than 30 marks?

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?

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.

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.

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.

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?

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

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

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

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?

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

Q. What thoughts or questions occur to you?

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.

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.

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.

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.

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

What could be the possible method to compile this data?

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.

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

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

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.

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?

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

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

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?

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

10. 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.

11. 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

(c) 12 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.

15. 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?