Tales by Dots and Lines | CBSE Class 8 Maths Notes

This note covers the arithmetic mean as a balance point, changes in mean and median, missing values, frequency tables, spreadsheet calculations, line graphs, temperature and rainfall patterns, space launches, infographics, activity strips and the interpretation of averages.

How does the mean balance a collection of values?

What do mean, median and dot plot mean?

Data are the values collected for a question or investigation. The arithmetic mean, also called the average here, is their sum divided by the number of values. The median is the middle value of sorted data, with the two middle values averaged when necessary.

A dot plot places a mark above each value on a number line. Repeated values have repeated marks. This makes the position of the data visible, including values that occur more than once. A measure of central tendency represents a centre of the data.

Definition: Mean = sum of all values ÷ number of values. The sign ÷ means division. Every occurrence of a repeated value contributes to both the sum and the count.

Property: The mean is a balance point

The total distance from values below the mean to the mean equals the total distance from values above it to the mean. A value at the mean contributes no distance. The mean is not always halfway between the smallest and largest values.

Worked example 1. Find the means of the pairs 3, 7 and 8, 9.

Answer: (3 + 7) ÷ 2 = 5, and (8 + 9) ÷ 2 = 8.5. In each pair the mean lies exactly halfway between the two numbers. Brackets group the addition before division.

Worked example 2. Show that 12 balances the values 10, 10, 11 and 17.

Answer: Their sum is 48, giving a mean of 48 ÷ 4 = 12. Below 12, the distances total 2 + 2 + 1 = 5. Above 12, the distance is also 5.

Two dots stand at 10, one at 11 and one at 17. The mean is marked at 12, with distances 2, 2 and 1 to its left balancing distance 5 to its right.

Reference: NCERT Class 8, page 104

There is just one such balance point. Moving the proposed centre away from the mean changes the two distance totals in opposite directions, so their equality is lost.

What happens when values are added to or removed from data?

Property: A new value pulls the mean towards it

Including a value greater than the existing mean increases the mean. Including a value smaller than it decreases the mean. The balance interpretation explains this: the new value introduces distance on one side, and the centre must move to restore equal totals.

Including a value equal to the mean leaves it unchanged. This also fits the fair-share interpretation, in which the total is shared equally among all the values. The additional value supplies exactly the amount needed for one more share at the existing mean.

Removing a value reverses the reasoning. Removing a value above the mean decreases the mean; removing one below it increases the mean. Removing a value equal to it leaves the mean unchanged, provided some data remain.

How can several changes preserve the mean?

When several values are included together, consider their combined effect. Their distances below and above the original mean may balance each other. Merely increasing the number of values does not determine whether the mean rises, falls or stays unchanged.

The same reasoning helps with changes to existing values. A fall in one value can be balanced by rises in other values. If the total and the number of values both stay unchanged, the mean stays unchanged.

Worked example 3. A group has mean weight 65.3 kilograms, written kg. During the next month, one person loses 2 kg and two people gain 1 kg each. What happens to the mean?

Answer: The total change is −2 + 1 + 1 = 0 kg, where − indicates a decrease. The number of people is unchanged, so their mean remains 65.3 kg.

This information does not settle the new median. To determine the middle position, we would need to know how the changed weights affect the ordering of the group.

How does changing every value affect the mean?

Property: Adding a fixed amount shifts the mean equally

If every value increases by the same amount, the mean increases by that amount. If every value decreases by the same amount, the mean decreases by that amount. The relative position of the mean within the collection stays the same.

Let n be the number of values, and let x₁, x₂, …, xₙ name those values. The subscripts identify individual values, and the dots mean that the sequence continues. Let a be their mean. Then a = (x₁ + x₂ + … + xₙ) ÷ n.

Adding 3 to every value adds 3n to the total. Here 3n means 3 multiplied by n. Dividing this new total by n gives a + 3. The count stays n because the existing values change; no extra observation is included.

Worked example 4. The measured mean height of 24 students is 150.2 centimetres, written cm. Each student's shoes add 1 cm. Find the mean height without shoes.

Answer: Every measurement must decrease by 1 cm. Therefore, the corrected mean is 150.2 − 1 = 149.2 cm. Measuring all the heights again is unnecessary when this common correction is known.

Property: Multiplying every value multiplies the mean equally

If every value is doubled, its mean also doubles. Multiplying every value by 5 makes the new mean 5a, meaning five times a. This follows because multiplying each value by 5 multiplies their total by 5 while leaving their number unchanged.

1. Start with the original total x₁ + x₂ + … + xₙ.
2. Multiply every value by 5, making the total 5(x₁ + x₂ + … + xₙ).
3. Divide by the unchanged number of values, n.
4. The original total divided by n is a, so the new mean is 5a.

The step that takes out the common multiplier uses the distributive property: multiplying a sum gives the same result as multiplying each term and then adding. Distinguish this operation from multiplying just one value.

How should the median be found and checked after a change?

Why must the data be sorted?

Sorted data are arranged in increasing or decreasing order. For a median calculation, increasing order makes it easy to count from the smallest value towards the middle. The median depends on position, so adding or removing a value requires checking the new ordered collection.

For an odd number of observations, one value occupies the middle position. For an even number, the median is the mean of the two middle values. Those middle values can be equal; repeated entries must retain their separate positions.

The first dot plot has values 4, 7, 8, 12 and 14, with median 8. Another plot adds a differently coloured dot at 11. The final plot marks the new median, 9.5, between 8 and 11.

Reference: NCERT Class 8, page 108

Worked example 5. Find the median of 4, 7, 8, 12 and 14, and then include 11.

Answer: The original middle value is 8. After including 11, the ordered values are 4, 7, 8, 11, 12, 14. The two middle values are 8 and 11, so the median becomes (8 + 11) ÷ 2 = 9.5.

Can a median remain unchanged?

Yes. Consider the ordered collection 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92. Its two middle values are both 41. Therefore, the median is 41.

Including another 41 preserves that middle value. Removing one of the two existing 41s also leaves a middle value of 41. These cases show why checking the actual middle positions matters more than assuming that every addition or removal must change the median.

How can a known mean reveal missing or incorrect data?

How do we recover the total?

The mean formula can be read backwards: total = mean × number of values, where × means multiplication. If one value is missing, subtract the sum of the known values from this total. Keep the number of observations separate from the sum of their measurements.

Worked example 6. Ten wrestlers have mean weight 39.2 kg. Nine weights are 42, 40, 39, 33, 48, 38, 42, 35 and 32 kg. Find the missing weight.

Answer: Let w be the missing weight in kg. The known weights total 349 kg. The required total is 39.2 × 10 = 392 kg. Thus 349 + w = 392, giving w = 392 − 349 = 43 kg.

Notice that the divisor is 10 because the average describes all ten players. Dividing the nine visible measurements by 9 answers a different question and cannot recover the smudged entry.

How is one mistaken entry corrected?

Worked example 7. Venkayya records an average coconut harvest of 25.6 per tree for 15 trees. One tree's count was recorded as 3 more than its actual harvest. Find the corrected average.

Answer: The recorded total is 25.6 × 15 = 384 coconuts. The corrected total is 384 − 3 = 381. Divide by the same 15 trees: 381 ÷ 15 = 25.4 coconuts per tree.

The correction belongs to one entry, so it is subtracted from the total once. It must not be subtracted directly from every tree's contribution or from the average. The number of trees is unchanged.

Worked example 8. A collection has 15 values with mean 134. Find its sum.

Answer: The sum is 134 × 15 = 2010. This uses the mean and count together; the individual values are not needed to recover their total.

How are mean and median calculated from frequencies?

What does a frequency count?

The frequency of a value is the number of times it occurs. A frequency table compresses repeated data without removing any observations. For family-size data, the value is the number of family members, while its frequency counts the students reporting that size.

Adding the eight different family sizes and dividing by 8 gives 6.5. That ignores how frequently each size occurs. There are 36 observations in this collection, rather than eight equally frequent observations.

Worked example 9. Find the mean family size using the frequency table.

Answer: Add the frequencies: 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1 = 36. Multiply each size by its frequency and add: 9 + 44 + 45 + 42 + 21 + 8 + 9 + 10 = 188. The mean is 188 ÷ 36, or 5.22 to two decimal places.

How do running frequency totals locate the median?

There is no need to write all 36 values. Add frequencies from the smallest family size upwards. The first three observations are 3. The next eleven are 4, taking the running total to 14. The next nine are 5, taking it to 23.

Worked example 10. Find the median family size from the same table.

Answer: With 36 observations, the middle positions are the 18th and 19th. Positions 15 to 23 all contain 5, so both middle values are 5. Their mean, and hence the median family size, is 5.

Position and value answer different questions. Eighteen and nineteen identify places in the ordered collection. They are not the family sizes to average. Similarly, a running frequency total tells us how far we have counted, not the value at every preceding position.

How can spreadsheets calculate totals and averages?

What are cells and ranges?

A spreadsheet is a digital arrangement of rows and columns containing small boxes called cells. Cells can contain text, numbers or formulae. Columns use letters and rows use numbers. A cell address gives the column first and the row second.

In Sudhakar's marks sheet, E5 means column E, row 5. It contains Farooq's Mathematics score. A range identifies a group of cells using its first and last addresses. The colon in B3:G3 means the cells from B3 through G3, including both ends.

Nagesh's marks occupy B3:G3. Gowri's Odia, Telugu and English marks occupy B7:D7. The range D2:D6 runs down one column and contains the English marks of the first five students. Reading the endpoints prevents selecting an unintended row or subject.

What do SUM and AVERAGE do?

A spreadsheet formula is an instruction that calculates a result. The initial equals sign tells the spreadsheet that a formula follows. SUM adds the numbers in the selected range; AVERAGE calculates their arithmetic mean.

Worked example 11. Nagesh's six subject marks in B3:G3 are 41, 43, 48, 39, 40 and 39. Find his total using a formula.

Answer: Enter =SUM(B3:G3). It adds 41 + 43 + 48 + 39 + 40 + 39 and returns 250.

Worked example 12. Gowri's marks in B7:D7 are 27, 29 and 34. Find their average using a formula.

Answer: Enter =AVERAGE(B7:D7). Their sum is 90, and there are 3 marks, so the result is 90 ÷ 3 = 30.

Row totals compare each student's combined marks, while column averages describe class performance in a subject. The calculation is quicker, but selecting the correct range remains essential to answering the intended question.

How do we construct and read a line graph?

What information does a line graph display?

A line graph connects plotted data points with line segments. It is generally used to visualise data across time. An axis is a reference line along which values are marked. The scale tells us what the spacing between marks represents.

The horizontal axis runs across the page; the vertical axis runs upwards. Read both labels before interpreting a point. A legend identifies the quantities or groups represented by different colours or markers. Markers are the shapes used to distinguish plotted points.

The following table gives average numbers of customers visiting a shop and purchasing items on each day. Both groups must be represented against the same sequence of days so that their patterns can be compared.

What steps make the graph readable?

1. Label the horizontal axis with the days in order from Monday to Sunday.
2. Label the vertical axis with the average number of customers and choose a consistent scale covering the table's values.
3. Plot the visiting values and connect consecutive points in their day order.
4. Plot the purchasing values, connect that set separately, and distinguish the two sets with a clear legend.

Use the seven daily pairs in the table. Draw separate connected lines for visiting and purchasing, with labelled axes and distinguishable markers.

Interpretation has two stages: first identify what is given, including the scale and organisation; then infer and interpret patterns. An inference is a conclusion drawn from the data. Knowing how the data were collected helps reveal missing information or possible bias, meaning a distortion in the evidence.

What do the Kerala and Punjab temperature graphs show?

Which temperature is being measured?

The graph compares monthly maximum temperatures in Kerala and Punjab during 2023. Maximum means the highest value. Temperature is shown in °C, meaning degrees Celsius. The horizontal axis lists months and the vertical axis gives temperature.

There could be a few weather stations across a state recording local temperatures regularly. The monthly maximum can be obtained by selecting the highest value among all those recorded across the state during the month. Understanding this measurement helps establish the graph's scope.

Kerala is represented by blue circular markers and blue lines; Punjab by red square markers and red lines. The months run from January to December. The vertical scale is temperature in degrees Celsius.

Reference: NCERT Class 8, page 117

How do the two patterns compare?

Punjab's monthly maximum rises from January to June, reaching 38°C. It falls to just under 35°C in July, stays mostly flat till September, and then falls continuously till December to about 23°C. January has the lowest monthly maximum, about 19°C.

Kerala's line stays mostly flat through the year. Its peak is around 33°C in April, and its lowest point is around 29°C in July. Its monthly maximum temperatures are similar in summer and winter.

The comparison shows greater temperature variation in Punjab. It also raises further questions about why the patterns differ, whether other states have similar patterns, and what their monthly minimum temperatures would show. Those questions call for additional evidence.

Different marker shapes help distinguish the lines when colours are difficult to distinguish or the graph is printed in black and white. Read the marker and legend together, rather than depending entirely on colour.

What can space-launch and rainfall graphs tell us?

How can trends and missing information be recognised?

A trend is a pattern of change across the data. The space-launch graph shows the annual, meaning yearly, number of objects launched into space. It includes lines for the world, the United States of America, China and Russia, across 2012 to 2024.

The horizontal axis gives years and the vertical axis gives the number of objects. Four labelled lines distinguish the world, the United States, China and Russia. The worldwide line falls slightly from 2023 to 2024.

Reference: NCERT Class 8, page 119

The worldwide count is around 2800 in 2024 and around 2900 in 2023. The claim that 2023 had the highest worldwide count carries the condition assuming the trend before 2012 was decreasing. Keep that condition when making the wider claim.

The three countries' counts do not add up to the worldwide count, so other countries are not individually displayed. A missing country line does not establish that the country launched nothing. Absence from a visualisation is different from a recorded zero.

For the United States, the increase from 2022 to 2023 exceeds the increase from 2023 to 2024. Over these equal time intervals on the same graph, the steeper segment represents the greater increase. A column version would need 13 groups of 4 bars, or 52 bars.

How are monthly rainfall averages interpreted?

Monthly average rainfall is obtained by averaging the total rainfall for the same month across several years. The rainfall graphs compare six cities. Rainfall is measured in millimetres, written mm.

Kovalam, Udupi and Mumbai lie along the west coast; Rameswaram, Chennai and Puri lie along the east coast. It appears that regions along the west coast receive more rain. Udupi, Mumbai and Kovalam have peak rainfall during June to August.

Rameswaram gets most of its rain during October to December. Chennai's rainfall begins from June onwards, peaks in November and continues till December. Puri peaks during July to September. January to March are dry months with low rainfall for all these cities; Rameswaram receives very little rain from January to September.

How do infographics and activity strips communicate data?

How should a colour scale be read?

An infographic combines visual features and information to communicate patterns clearly and quickly. Its colours, labels and scale must be interpreted together. A colour does not supply a complete meaning without the key that explains what it represents.

The rice-and-wheat infographic maps differences in per capita consumption, meaning consumption per person. These differences are represented on a scale from −100 to +100. The minus and plus signs identify the two sides of the preference scale.

A map uses yellow-to-purple shading and a scale labelled from mostly wheat to mostly rice. A red line highlights a geographical split in preferences. The map is marked as not to scale.

Reference: NCERT Class 8, page 124

A value of +100 does not mean that no wheat is consumed. It indicates mostly rice and the highest difference between per capita rice and wheat consumption. Reading the mapped quantity correctly prevents turning a preference into a claim of complete exclusion.

How does a strip represent one day?

An activity strip divides a day into time intervals and colours them by activity. Manoj uses 48 boxes, each representing 30 minutes, from midnight to midnight. Three strips record three different days and allow their routines to be compared.

The listed activities include sleeping; eating; meeting friends, hobbies, media and family time; classes, studying and homework; showering, dressing, yoga or exercise; and travelling. A run of boxes shows when an activity happens and how long it lasts.

For a personal investigation, record activities on different days, calculate the average time spent on each, and represent an average day with a strip. Comparing a student's strip with an adult's can raise questions about how daily routines differ.

Note: A strip showing an average day summarises time allocations. Keep the original daily strips when comparing the timing and regularity of eating, sleeping and other activities.

How can average sleep graphs lead to better questions?

What pattern is visible across ages?

The sleep graphs show typical sleep durations of Indians from ages 6 to 75. The horizontal axis gives age in years; the vertical axis gives sleep in hours per day. The second graph is a zoomed-in version of the first, showing the pattern with a different vertical scale.

Two blue curves fall from younger ages, flatten through the middle ages and rise again at older ages. The second panel uses a narrower vertical scale, making the changes appear more pronounced.

Reference: NCERT Class 8, page 126

The average sleep time for 6-year-olds is about 9.5 hours per day. Daily sleep time decreases through the teenage years towards adulthood, touching about 8 hours between ages 30 and 50. After 50, it increases, reaching about 8.5 hours.

Typical or average duration is a summary of a group. Sleep duration can vary greatly among people. Sleep needs depend on age, living conditions and lifestyle, including food and activities, among other factors. The graph does not assign an identical duration to every person of one age.

How can data collection extend the investigation?

Closely spaced points can make a line graph look like a smooth curve. Changing the displayed scale can alter the visual impression, so compare the numbers on the axes before deciding how large a variation is.

A sleep investigation can track family members for a week, counting night sleep, naps and other daytime sleep. Represent the observations on strips, then calculate mean and median sleep times for children, adults and elderly people.

The purpose of interpretation is both to describe the available evidence and to identify useful new questions. Whether people in different countries have similar sleep patterns requires more data; a graph for Indians alone cannot settle that comparison.

Glossary

• Arithmetic mean — The sum of every value divided by the number of values in the collection.
• Median — The middle of ordered data, averaging the two middle values when the count is even.
• Dot plot — A display placing a separate mark above each observed value on a number line.
• Frequency — The number of times a particular value occurs in a collection of data.
• Fair share — The amount each would receive if a total were shared equally among all observations.
• Spreadsheet — A digital arrangement of cells in rows and columns for recording data and performing calculations.
• Cell address — The column letter followed by the row number identifying one box in a spreadsheet.
• Range — A group of spreadsheet cells identified by its starting and ending cell addresses.
• Line graph — A graph joining plotted points with line segments, generally used to visualise data across time.
• Scale — The relationship between marked positions on a graph and the values they represent.
• Legend — A key explaining which data groups are represented by different colours or marker shapes.
• Inference — A conclusion drawn by reasoning from the information available in data or a graph.
• Infographic — A visual presentation combining information and graphical features to communicate patterns clearly and quickly.
• Per capita consumption — Consumption expressed per person, used when comparing consumption levels across different groups or places.
• Activity strip — A sequence of coloured time intervals showing activities and their durations through a day.

Common errors and misconceptions

• Misconception: The mean is halfway between the smallest and largest values. Correct: That is not always so. The mean balances total distances on either side, counting repeated values separately.
• Misconception: Increasing the number of values must increase the mean. Correct: The new values' sizes relative to the existing mean determine the change, rather than their number alone.
• Misconception: Correcting one entry by 3 means subtracting 3 from the mean. Correct: Correct the total by 3, then divide by the unchanged number of observations.
• Misconception: A frequency-table mean uses each different value once. Correct: Multiply each value by its frequency, add these products, and divide by the total frequency.
• Misconception: The median is the mean of the two middle position numbers. Correct: Average the values occupying those positions, after arranging the complete collection in order.
• Misconception: A country absent from a launch graph must have launched nothing. Correct: The graph may omit that country's separate count; missing information is not a recorded zero.
• Misconception: A rice-preference value of +100 means no wheat consumption. Correct: It indicates mostly rice and the highest difference between per capita rice and wheat consumption.
• Misconception: Everyone of the same age sleeps for the plotted average duration. Correct: A group average summarises data; individual sleep durations can vary greatly.

Exam-style questions with model answers

Q1. Find the mean of 3 and 7. Explain its position relative to the two numbers. [2 marks]

1. The arithmetic mean is their sum divided by their count: (3 + 7) ÷ 2 = 5.
2. Five lies exactly halfway between 3 and 7, at a distance of 2 from each number.

Q2. The mean measured height of 24 students is 150.2 cm. Each student's shoes add exactly 1 cm. Find the mean height without shoes and explain why remeasurement is unnecessary. [3 marks]

1. Each recorded height includes the same additional 1 cm from the shoes, so each value needs the same subtraction.
2. Subtracting a fixed amount from every value subtracts that amount from the mean. Therefore, the corrected mean is 150.2 − 1 = 149.2 cm.
3. Remeasurement is unnecessary because the common correction is known and the same 24 students remain in the collection.

Q3. Ten wrestlers have mean weight 39.2 kg. Nine weights are 42, 40, 39, 33, 48, 38, 42, 35 and 32 kg. Find the missing weight, showing the total and the sum of the known weights. [4 marks]

1. The mean describes all ten wrestlers, so their combined weight equals the mean multiplied by ten.
2. The required total weight is 39.2 × 10 = 392 kg.
3. The nine known weights add to 42 + 40 + 39 + 33 + 48 + 38 + 42 + 35 + 32 = 349 kg.
4. The missing weight is the difference between these totals: 392 − 349 = 43 kg.

Q4. A farm has 15 coconut trees with recorded average harvest 25.6 coconuts per tree. One tree's harvest was recorded as 3 more than the actual count. Find the corrected average and explain why 3 is not subtracted directly from 25.6. [4 marks]

1. The recorded average and tree count give a recorded total of 25.6 × 15 = 384 coconuts.
2. The error is an overcount of 3 in one entry, so the corrected total is 384 − 3 = 381 coconuts.
3. The number of trees stays 15. The corrected average is therefore 381 ÷ 15 = 25.4 coconuts per tree.
4. Subtracting 3 directly from the average would incorrectly apply that correction to every tree's share, rather than once to the total.

Q5. Family sizes 3, 4, 5, 6, 7, 8, 9 and 10 have respective frequencies 3, 11, 9, 7, 3, 1, 1 and 1. Calculate the mean to two decimal places and the median, explaining how frequencies locate the middle values. [5 marks]

1. The number of observations is the sum of all frequencies: 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1 = 36.
2. Counting each family size according to its frequency gives the total 9 + 44 + 45 + 42 + 21 + 8 + 9 + 10 = 188.
3. Divide this total by the number of observations. The mean family size is 188 ÷ 36, which is 5.22 to two decimal places.
4. There are 36 ordered values, so the median uses the 18th and 19th positions. The frequencies of sizes 3 and 4 together cover the first 14 positions.
5. The nine occurrences of size 5 occupy positions 15 to 23. Both middle values are therefore 5, and their mean gives a median of 5.

Q6. A spreadsheet contains Nagesh's six marks, 41, 43, 48, 39, 40 and 39, in cells B3:G3. Explain the range notation, give the SUM formula and calculate the result. [3 marks]

1. B3:G3 means the cells in row 3 from column B through column G, including both endpoints. The colon joins the starting and ending cell addresses.
2. Enter =SUM(B3:G3). The initial equals sign introduces the formula, and SUM adds the numbers in the specified range.
3. The total is 41 + 43 + 48 + 39 + 40 + 39 = 250 marks.

Q7. A 2023 monthly maximum-temperature graph shows Punjab rising from about 19°C in January to 38°C in June, falling to just under 35°C in July, staying mostly flat till September, then falling to about 23°C in December. Kerala stays mostly flat, peaking around 33°C in April and reaching its lowest point of around 29°C in July. Identify suitable axes and compare the two patterns, keeping the stated approximations. [5 marks]

1. The horizontal axis should show the months from January to December. The vertical axis should show monthly maximum temperature in degrees Celsius.
2. Punjab rises from about 19°C in January to 38°C in June. June therefore has the highest monthly maximum described for Punjab.
3. Punjab falls to just under 35°C in July, stays mostly flat till September, and then falls to about 23°C in December.
4. Kerala stays mostly flat through the year, with a peak around 33°C in April and its lowest point around 29°C in July.
5. Punjab shows greater variation than Kerala. Kerala's monthly maximum temperatures are similar in summer and winter, while Punjab's pattern changes much more across the year.

Q8. A group's mean weight is 65.3 kg. Next month, the same group has one person losing 2 kg and two people gaining 1 kg each; all other weights are unchanged. Find the new mean and explain. [2 marks]

1. The total weight change is −2 + 1 + 1 = 0 kg, so the combined weight remains unchanged.
2. The number of people also remains unchanged. Dividing the same total by the same count gives the same mean, 65.3 kg.

Key takeaways

• The arithmetic mean balances the total distances of all observations below and above it, including repeated values separately.
• A value above the existing mean raises it when included; a value below it lowers it.
• Adding the same amount to every observation adds that amount to the mean without changing the observation count.
• Multiply each value by its frequency before adding when calculating a mean from a frequency table.
• The median uses ordered positions; running frequency totals can locate those positions without writing every repeated value.
• Spreadsheet ranges identify the cells to calculate with, while SUM and AVERAGE perform different operations on their numbers.
• Read graph labels, scales, markers and collection methods before interpreting trends or drawing conclusions from the data.
• Keep approximate readings and conditional conclusions qualified, and distinguish group averages from claims about every individual.

Test yourself

Why is 12 the balance point for 10, 10, 11 and 17?

The distances below 12 total 2 + 2 + 1 = 5, equal to the distance 5 above it.

What happens to the mean when a value equal to it is included?

The mean stays unchanged because the new value supplies exactly one more share at the existing mean.

If every value is multiplied by 5, what happens to its mean?

The mean is multiplied by 5 because the total is multiplied by 5 and the count stays unchanged.

A collection contains 15 values with mean 134. What is its sum?

Multiply the mean by the number of values: 134 × 15 = 2010.

Why must repeated values be counted in a mean calculation?

Each repeated value is a separate observation and contributes to both the total and the number of values.

What does D2:D6 mean in a spreadsheet?

It identifies the cells in column D from row 2 through row 6, including both endpoints.

Does a value of +100 on the rice-preference scale mean that no wheat is eaten?

No. It indicates mostly rice and the highest difference between per capita rice and wheat consumption.

How much time does one box in Manoj's 48-box daily strip represent?

Each box represents 30 minutes, with the complete strip extending from midnight to midnight.