Connecting the Dots... | CBSE Class 7 Maths Notes

This note covers statistical questions and statements, representative values, arithmetic mean, equal sharing, dot plots, median, outliers, variability, zero and missing values, clustered bar graphs, and drawing conclusions from data.

What makes a question or statement statistical?

Data are the values or information collected to investigate a question. Statistics is the study of collecting, organising, analysing, interpreting, and presenting data. Its purpose includes finding patterns and explaining what a collection of values tells us.

Definition: A statistical question can be answered by collecting data. A statistical statement is a claim or summary about something, expressed through numerical values, proportions, probabilities, or predictions.

A proportion describes a part in relation to a whole. A probability describes chance, while a prediction expresses what is expected to happen. A statistical statement need not be a certainty merely because it contains a number.

How does variation shape a question?

“How tall are Grade 7 students in our school?” is a statistical question. We expect different heights. Collecting and examining those heights helps us describe the class without assuming that every student has the same height.

Asking whether onions are typically costlier in Yahapur or Wahapur also requires data. Prices can vary over time. One price observed in one month does not describe all the prices experienced during a year.

Statistical thinking connects observations with conclusions while allowing for uncertainty. A claim that a journey takes about 15 minutes expresses an approximate duration. The word “about” matters: removing it changes an approximate statement into an exact one.

After collecting data, organise it so that comparisons become easier. Then explain what the data supports and what remains uncertain. This sequence turns a question into an investigation and makes it possible to ask useful further questions.

Why can different representative values give different comparisons?

A representative value is a number used to describe a collection of values. The total is their sum, the minimum is the smallest value, and the maximum is the largest. Each highlights a different feature.

Consider the runs scored by Shubman and Yashasvi in a cricket series. Comparing a single match, the highest score, or the whole series can lead to different impressions of their performances.

Yashasvi scored more in the first two matches; Shubman scored more in the last two. Shubman achieved the highest individual score, 90. However, Yashasvi's total was 175, compared with Shubman's 128.

Result: Range measures the gap between the extremes

The range is the difference between the maximum and minimum values. It helps describe variability, meaning how the values differ or spread out. Here, Yashasvi's smaller difference between highest and lowest scores indicates more consistent batting.

Worked example 1. Use the four scores in the table to compare the players' ranges.

Answer: Shubman's range is 90 − 0 = 90 runs. Yashasvi's range is 67 − 18 = 49 runs. The sign − means subtraction; = means equality. Yashasvi has the smaller range.

It is often not simple to compare two groups of numbers and clearly say that one is better. If the groups contain different numbers of values, their totals may not be appropriate for comparison. The question being investigated helps determine which representative value is useful.

How does the arithmetic mean express an equal share?

The arithmetic mean, also called the mean or average, balances the high and low values in a collection. Add all the values and divide by the number of values. The result describes what each share would be if the total were distributed equally.

Result: Mean is the total divided by the count

Definition: Mean = sum of all values ÷ number of values. The sign ÷ means division. The number of values is the count of observations included in the calculation.

Equal sharing explains why both parts of the calculation matter. Two groups may collect the same total but receive different shares if one group has more members. Keeping the total unchanged does not keep the mean unchanged when the group size changes.

Worked example 2. Shreyas and four friends collect 3, 8, 10, 5, and 4 guavas. Parag and five friends collect 5, 4, 6, 3, 4, and 8 guavas. Compare their equal shares.

Answer: The sign + means addition. Shreyas's group collects 3 + 8 + 10 + 5 + 4 = 30 guavas, shared among five people: 30 ÷ 5 = 6. Parag's group also collects 30 guavas, shared among six people: 30 ÷ 6 = 5. Each member of Shreyas's group gets one more guava.

Does an average describe every actual observation?

Worked example 3. Vaishnavi records 2, 7, 9, 4, and 3 hibiscus flowers blooming over five days. Find the average daily number.

Answer: The total is 2 + 7 + 9 + 4 + 3 = 25 flowers. The mean is 25 ÷ 5 = 5 flowers per day. This is the daily count if an equal number bloomed each day.

The five actual flower counts differ from one another and from the mean. Thus, “five flowers per day on average” does not say that five flowers actually bloomed on every day. It summarises the collection through an equal-share value.

How do we calculate a mean carefully?

Before calculating a mean, identify what each value measures and how many observations belong in the calculation. A list of daily flower counts, a list of cricket scores, and a list of attempts at bouncing a ball all require the same basic operation.

What steps keep the calculation clear?

1. Read the question and identify the collection whose mean is required.
2. Count the recorded observations, including actual zero values where they belong.
3. Add all the values in that collection to obtain the total.
4. Divide the total by the number of observations and interpret the answer in context.

Worked example 4. Shreyas bounces a ball on a bat. The numbers of bounces in eight attempts are 6, 2, 9, 5, 4, 6, 3, and 5. Find his mean number of bounces.

Answer: The total is 6 + 2 + 9 + 5 + 4 + 6 + 3 + 5 = 40 bounces. There are eight attempts, so the mean is 40 ÷ 8 = 5 bounces per attempt.

Repeated values still represent separate observations. The two attempts with six bounces both contribute to the total and the count. Likewise, both attempts with five bounces must be included. Counting only distinct values would change the collection being studied.

The meaning of the measurement matters when comparing means. For the running times of two friends covering the same 100 metres, a lower mean time indicates quicker running on average. For runs scored per match, a higher mean describes more runs per match.

A numerical answer should therefore be followed by a sentence explaining it. Calculation produces a representative number; interpretation connects that number to the question.

What do dot plots reveal about onion prices?

A dot plot represents each data value by a dot positioned along a number line. Repeated values are stacked at the same position. This makes their frequency, or number of occurrences, visible while also showing where values gather or spread out.

The monthly onion prices below are in rupees per kilogram. The symbol ₹ means rupees, and kg means kilogram. Reading across a row compares the two towns in the same month.

How is a dot plot read?

Green dots represent Yahapur and purple diamond-shaped dots represent Wahapur. Both horizontal number lines run from 10 to 60. Two purple dots are stacked at 42, showing two occurrences of that price.

Reference: NCERT Class 7, page 103

The spacing between equal numerical intervals must be equal. The plot need not begin at zero here because the values lie above 10 and do not exceed 60. The ends reveal the minimum and maximum; the arrangement reveals the spread.

Wahapur's prices are more spread out. Its minimum is 17 and maximum is 60, giving a range of 43. Yahapur's minimum is 24 and maximum is 59, giving a range of 35. These compare variation, rather than just the highest price.

What information is lost?

A dot plot sorts the prices by value but loses their original month-wise sequence. It cannot identify which Yahapur dot belongs to January. Use the table to recover that link. Choose a display according to whether the question concerns distribution or change over time.

The table also allows different comparisons: Yahapur has higher prices in six months, Wahapur in five, and their prices match in one. A single largest price cannot replace this month-by-month description.

How is the median found for odd and even numbers of values?

The median is the middle value of sorted data. Sorted data means values arranged in order, such as from smallest to largest. Sorting is essential: the value halfway along an unsorted list need not be the median.

Result: The median depends on the middle position or positions

With an odd number of observations, meaning a count not divisible into two equal whole-number groups, select the single middle value. With an even count, meaning one divisible into two equal whole-number groups, average the two middle values.

Worked example 5. Poovizhi's family heights are 170, 173, 165, 118, and 175 centimetres. The abbreviation cm means centimetre. Find the median height.

Answer: Sort the heights as 118, 165, 170, 173, 175. There are five values. The third value is in the middle, with two positions before it and two after it. The median is 170 cm.

For an even count, there is no single middle observation. Taking the mean of the two middle values gives the median. This calculation concerns those two values, not the mean of the entire collection.

Worked example 6. Yaangba's family heights are 169, 173, 155, 165, 160, and 164 cm. Find the median height.

Answer: Sorting gives 155, 160, 164, 165, 169, 173. The two middle values are 164 and 165. Their mean is (164 + 165) ÷ 2 = 164.5 cm. Parentheses group the addition to be completed before division.

Mean and median are measures of central tendency. Central tendency describes the tendency of values to pile up around a particular value: the measures represent a centre of the data. They can give different perspectives on the same collection.

Use positions to identify the middle, particularly when values repeat. Repeated observations still occupy separate positions in the sorted list, just as they contribute separately when calculating a mean.

How can outliers affect the mean and median?

An outlier is a value that differs significantly from the rest of the data. A very low or very high value can change the total substantially and therefore affect the mean. The average may not always be an appropriate representative of data containing outliers.

In Poovizhi's family, 118 cm is much lower than the other heights, 170, 173, 165, and 175 cm. The mean height is 160.2 cm, which is below the heights of four of the five family members.

What does the family comparison show?

Although most members in Poovizhi's family are taller, their family's average is below Yaangba's family average of 164.3 cm. Here, the average does not seem to represent Poovizhi's family heights very well. The median of 170 cm offers another perspective.

The dot plots mark individual heights and vertical lines for the means. The lower pair also marks the medians with dashed lines. Poovizhi's 118 cm point sits apart from the other four heights.

Reference: NCERT Class 7, page 106

Worked example 7. Investigate Poovizhi's heights after leaving out the outlier 118 cm. The remaining values are 170, 173, 165, and 175 cm.

Answer: Their total is 683 cm, so the mean is 683 ÷ 4 = 170.75 cm. Sorting gives 165, 170, 173, 175. The median is (170 + 173) ÷ 2 = 171.5 cm. Compare these with the original mean 160.2 cm and median 170 cm.

This investigation shows why an outlier's influence deserves attention; it is not a direction to discard observations automatically. In these family examples, the median was not affected much by the outlier. The mean changed much more.

When data is more balanced or uniformly spread out, the mean and median appear to be close. With an outlier at the lower end, the mean appears to shift downwards; with one at the higher end, it appears to shift upwards.

Why must zero be distinguished from a missing value?

A zero value records an actual result of zero. A missing value means that no value has been recorded for an observation. In cricket, scoring zero in a match differs from not playing that match.

Suppose a player's scores are 57, 13, 0, 84, no appearance, 51, and 27. The player took part in six matches. The zero belongs in the data; the match not played does not become an extra zero score.

Worked example 8. Find the mean runs per match for the six scores 57, 13, 0, 84, 51, and 27.

Answer: The total is 57 + 13 + 0 + 84 + 51 + 27 = 232 runs. Divide by six matches: 232 ÷ 6 = 38⅔ runs per match. The mixed number 38⅔ means 38 plus two-thirds.

How does the question determine which observations belong?

For a class survey of animals kept at home, an absent student's missing response is different from a response of zero animals. Decide which responses have actually been collected before finding the mean or median.

Context also matters when the collection concerns seasonal events. Sita's mango tree produces fruit in some months and none in others. To study the typical monthly production during the growing season, it would be appropriate to consider the summer months when mangoes are expected to grow.

This is a choice about the period being investigated, not a general rule to remove zeros. First establish the question and relevant period. Then distinguish real zero results within that period from values that were not observed or recorded.

How should we compare the centre and spread together?

A centre helps summarise typical values, while the spread shows how widely values differ. To describe a collection, consider its minimum, maximum, range, mean, and median together. A dot plot makes this combined description easier to see.

What do newspaper page counts show?

Worked example 9. A newspaper has 16, 18, 20, 22, 26, 16, and 10 pages from Monday to Sunday. Find its mean and median page count.

Answer: The total is 128 pages across seven days, giving a mean of 128 ÷ 7 pages, approximately 18.29 pages. Sorting gives 10, 16, 16, 18, 20, 22, 26. The fourth value is the median, 18 pages.

The mean and median are close in this example. Describing the minimum of 10 and maximum of 26 adds information that the centre alone does not provide. The repeated count of 16 remains visible as two observations on a dot plot.

Does a comparison of averages describe every individual?

In the Grade 5 class height example, boys' heights range from 128 to 158 cm, while girls' heights range from 136 to 156 cm. Both the tallest and shortest students are boys, yet the girls have a higher average height.

Saying girls are taller than boys in this class does not mean that every girl is taller than every boy. A statement about a group's average describes the group through a representative value. It must not be converted into a claim about every member.

Look for both differences and overlap when comparing groups. The location of the centre, the two extremes, and the pattern of the dots answer different parts of the question. None of these descriptions alone gives the whole picture.

How do clustered bar graphs make comparisons easier?

A bar graph represents values using the lengths or heights of bars. A clustered bar graph places related bars side by side for comparison. With two vertical columns in each cluster, it is also called a double column graph.

The onion-price data can be shown as two separate column graphs, but placing the towns' bars beside one another helps compare each month directly. A scale states how numerical values correspond to distances along a graph's numbered line.

Each month has two adjacent bars. Blue bars with slanted white lines show Yahapur; red bars with white dots show Wahapur. The vertical price markings rise from 0 to 60 in steps of 10.

Reference: NCERT Class 7, page 115

How should the graph be constructed?

1. List the months in their January-to-December order along the horizontal line.
2. Mark a suitable, evenly spaced price scale along the vertical line.
3. Draw the two towns' bars next to each other for each month, using the table values.
4. Include a key identifying each town, with distinguishable colours or patterns.

A key explains which colour or pattern represents each collection. Patterns help people who have difficulty distinguishing colours and also help when the graph is printed in black and white. They communicate the same distinction as the colours.

Compare bar heights within a month to identify the costlier town. Follow one town's bars across the months to examine its changes over time. Unlike the onion dot plots, this display preserves the link between each price and its month.

For months, chronological order is meaningful. For categories such as different rocket-launching organisations, changing the category order does not change the underlying comparison. Always identify what the categories represent before interpreting their arrangement.

How can we separate graph observations from interpretations?

Use two stages when a graph contains many values: identify what is given, then infer from what is given. To infer is to draw a conclusion from observations. Begin with the title, categories, scale, key, and patterns before writing a summary.

What does the rocket-launch graph illustrate?

The rocket-launch graph groups three adjacent bars for each organisation, covering 2021, 2022, and 2023. Its numbered scale advances by 20 rockets per unit length. The “Others” category combines several organisations to keep the display shorter.

SpaceX launched about twice as many rockets in 2022 as in 2021, then about 35 more in 2023 than in 2022. Arianespace's launches decreased each year. United Launch Alliance's count rose in 2022 and then fell below both previous years in 2023.

These descriptions preserve the precision available from the graph. A reading labelled “Perhaps 61” remains an estimate; it should not become a definite exact count. Similarly, “around 25” for the 2023 Others category communicates an approximate reading.

What do daylight comparisons illustrate?

Daylight hours are hours when the Sun is at least partly above the horizon. To find average daylight hours per day in a month, divide that month's total daylight hours by the number of days in that month.

City 1 is Helsinki in Finland and City 2 is Wellington in New Zealand. Their daylight patterns run in opposite directions. City 1's average rises towards June and falls towards December; City 2's average falls towards June and rises towards December.

City 2 reaches a minimum of about 9 hours in June and a maximum of about 15 hours in December. City 1 has more extreme maximum and minimum values. These describe both the seasonal pattern and the difference in variability.

After identifying a pattern, ask what could help explain it. Further questions about location or earlier years extend an investigation; they are not answers already supplied by the displayed bars.

How does a data detective avoid unsupported conclusions?

A data detective looks for patterns, checks the meaning of the measurements, and asks what further information is needed. Well-organised data can suggest explanations and new questions, but conclusions must remain within what the observations can support.

When is a generalisation too broad?

Looking at heights in one or two schools cannot justify a conclusion about all children in the country or around the world. Even within one school, a difference between group averages does not mean every individual follows the same pattern.

Similarly, knowing the mean height of one section does not determine the mean of another section simply because both have the same numbers of boys and girls. Equal group sizes do not supply the actual heights needed for the second mean.

A tall family member bends at a low doorway while shorter family members stand nearby. The speech bubble says, “Perhaps using the average height of the family to make the door was not a good idea!”

Reference: NCERT Class 7, page 129

The cartoon prompts a practical question about using an average. A representative height summarises a collection; it does not automatically meet the needs of its tallest member. A doorway based on the average height may be too low for the tallest family member.

How can an investigation be extended?

Possible projects include comparing sentence lengths on pages from two subject textbooks, investigating the lengths of classmates' names, or collecting family heights. Record the actual observations, make dot plots, and compare their variability and central tendency.

Estimating and then measuring object lengths offers another investigation. Compare each estimate with its measurement and examine the positive differences. Here, a positive difference measures how far apart the two lengths are, using the larger minus the smaller.

Do not invent project results before collecting data. The purpose is to let observations guide the description and to notice new questions that arise from the evidence.

Glossary

• Data — Values or information collected to investigate questions and describe the observations made.
• Statistics — The study of collecting, organising, analysing, interpreting, and presenting data.
• Statistical question — A question answered by collecting data and examining the values obtained.
• Statistical statement — A claim or summary expressed through numerical values, proportions, probabilities, or predictions.
• Representative value — A number used to describe a collection of data values.
• Arithmetic mean — The sum of all data values divided by the number of values.
• Median — The middle value of sorted data, or the average of its two middle values.
• Outlier — A value that differs significantly from the rest of the observations in a collection.
• Central tendency — The tendency of data values to pile up around a particular value.
• Variability — The way values differ and are clustered or spread out within a collection.
• Range — The difference between the maximum and minimum values in a collection of data.
• Dot plot — A display placing a dot at the position of each data value along a line.
• Frequency — The number of times a particular value occurs in the data.
• Clustered bar graph — A graph placing related bars together to compare values across categories or time.
• Missing value — An observation for which no value is recorded, distinct from an actual recorded zero.

Common errors and misconceptions

• Misconception: The larger total proves better performance even when players have played different numbers of matches. Correct: Compare the total alongside the number of matches; mean runs per match accounts for that count.
• Misconception: An average of five flowers means five bloomed every day. Correct: The mean represents an equal daily share of the total; the actual counts can differ from it.
• Misconception: The median is the middle entry in the original list. Correct: Sort the data first. For an even number of observations, average the two middle values.
• Misconception: A missing cricket score counts as zero. Correct: A score of zero records a match played; a match not played is excluded when calculating runs per match played.
• Misconception: The mean represents data with outliers equally well in every situation. Correct: An extreme value can affect the total substantially; inspect the median and the distribution too.
• Misconception: An onion-price dot plot identifies each month's price. Correct: It displays values and their frequencies but loses the original month-wise order; consult the table for the month.
• Misconception: A higher average height means every member of that group is taller. Correct: An average compares groups through a representative value and does not establish every individual comparison.
• Misconception: A rough bar-graph estimate should be reported as an exact number. Correct: Keep qualifying words such as “about”, “around”, and “Perhaps” when the reading is approximate.

Exam-style questions with model answers

Q1. Explain why “How tall are Grade 7 students in our school?” is a statistical question. [2 marks]

1. The question can be answered by collecting students' heights as data.
2. The heights are expected to vary, so examining the collection helps describe the heights that occur.

Q2. Vaishnavi records 2, 7, 9, 4, and 3 hibiscus flowers over five days. Find the mean and explain its equal-share meaning. [3 marks]

1. Add the five daily observations: 2 + 7 + 9 + 4 + 3 = 25 flowers in total.
2. Divide the total by the five days observed. The mean is 25 ÷ 5 = 5 flowers per day.
3. This means five flowers would bloom daily if the same total were spread equally across those days; it does not say that each actual daily count was five.

Q3. Shreyas records 6, 2, 9, 5, 4, 6, 3, and 5 bounces in eight attempts. Calculate the mean and median, showing the total and sorted data. [4 marks]

1. The total number of bounces is 6 + 2 + 9 + 5 + 4 + 6 + 3 + 5 = 40.
2. There are eight recorded attempts, so the mean is 40 ÷ 8 = 5 bounces per attempt.
3. Arrange all the observations in increasing order: 2, 3, 4, 5, 5, 6, 6, 9. Repeated values retain separate positions.
4. The fourth and fifth observations are both 5. Their average, (5 + 5) ÷ 2 = 5, is the median number of bounces.

Q4. Poovizhi's family heights are 170, 173, 165, 118, and 175 cm. Find the mean and median, identify the outlier, and explain why the mean may represent this family poorly. [5 marks]

1. The total height is 170 + 173 + 165 + 118 + 175 = 801 cm. There are five family members in this collection.
2. The arithmetic mean is the total divided by the count: 801 ÷ 5 = 160.2 cm.
3. The sorted heights are 118, 165, 170, 173, 175. The third value is the single middle observation, making the median 170 cm.
4. The height 118 cm is an outlier because it differs significantly from the other four heights, which are much closer together.
5. The mean is lower than four of the five heights. The low outlier reduces the total and pulls the mean down, so the mean does not seem to represent these heights very well.

Q5. The twelve monthly onion prices in rupees per kg are Yahapur: 25, 24, 26, 28, 30, 35, 39, 43, 49, 56, 59, 44; Wahapur: 19, 17, 23, 30, 38, 35, 42, 39, 53, 60, 52, 42. Compare their ranges and medians and explain what these comparisons describe. [5 marks]

1. Yahapur's minimum is 24 and its maximum is 59. Its range is therefore 59 − 24 = 35 rupees per kg.
2. Wahapur's minimum is 17 and its maximum is 60. Its range is 60 − 17 = 43 rupees per kg, larger than Yahapur's range.
3. Yahapur's sorted prices are 24, 25, 26, 28, 30, 35, 39, 43, 44, 49, 56, 59. Its median is (35 + 39) ÷ 2 = 37 rupees per kg.
4. Wahapur's sorted prices are 17, 19, 23, 30, 35, 38, 39, 42, 42, 52, 53, 60. Its median is (38 + 39) ÷ 2 = 38.5 rupees per kg.
5. The ranges describe the gaps between extreme prices; Wahapur has the wider gap. The medians compare the centres of the sorted prices, with Wahapur having the higher median.

Q6. A player's results over seven scheduled matches are 57, 13, 0, 84, did not play, 51, and 27 runs. State the correct divisor, calculate mean runs per match played, and explain how zero differs from not playing. [3 marks]

1. The correct divisor is six, because the player took part in six matches. The scheduled match not played supplies no score.
2. The recorded scores total 57 + 13 + 0 + 84 + 51 + 27 = 232 runs. Mean runs per match played are 232 ÷ 6 = 38⅔.
3. The zero records an actual performance and remains in the count. Not playing does not create a seventh performance or an extra zero score.

Q7. A newspaper has 16, 18, 20, 22, 26, 16, and 10 pages over seven days. Find its mean, median, and range, and explain what each contributes to the description. [4 marks]

1. The total is 128 pages, so the mean is 128 ÷ 7, approximately 18.29 pages per day. It gives the equal-share daily page count.
2. Sorting gives 10, 16, 16, 18, 20, 22, 26. The fourth value is 18, so the median describes the middle of the ordered counts.
3. The minimum is 10 and maximum is 26. The range is 26 − 10 = 16 pages, describing the gap between the extremes.
4. The mean and median are close here, while the range adds information about variation that either central value alone would leave out.

Q8. Explain two advantages of showing two towns' monthly onion prices as adjacent bars with different patterns in a double column graph. [2 marks]

1. Adjacent bars make the towns' prices directly comparable within each month.
2. Different patterns distinguish the towns when colours are difficult to tell apart or the graph is printed in black and white.

Key takeaways

• Statistical questions are answered by collecting and examining data, while statistical statements summarise observations or express predictions.
• The arithmetic mean divides the total by the number of observations and represents an equal share of that total.
• Find the median after sorting every observation; with an even count, average the two middle values.
• Dot plots reveal repeated values, clusters, and spread, but the onion-price plots lose the original order of months.
• A very high or low outlier can affect the mean substantially; inspect the median and the distribution before summarising.
• Keep actual zeros distinct from missing values and decide which observations belong to the question being investigated.
• Read the categories, key, and scale of a clustered bar graph before comparing values or interpreting its patterns.
• Compare centres and extremes together, preserve approximate wording, and avoid turning group averages into claims about every individual.

Test yourself

What two quantities are needed to calculate the arithmetic mean?

The sum of all the observations and the number of observations included in that collection.

What is the first step in finding a median from an unordered list?

Sort all the observations into numerical order, retaining each repeated value as a separate observation.

In the sorted heights 155, 160, 164, 165, 169, 173 cm, which values determine the median?

The middle values are 164 and 165 cm; their average gives the median of 164.5 cm.

What do two stacked dots at 42 on Wahapur's onion-price plot represent?

They represent two separate monthly observations with the same price of 42 rupees per kilogram.

Can a dot plot of monthly prices identify January's price without month labels?

No. The values are arranged by price, so their original month-wise sequence has been lost.

Why does the 118 cm height in Poovizhi's family deserve attention when calculating the mean?

It is much lower than the other family heights and pulls the total, and therefore the mean, downwards.

What is the difference between a zero cricket score and a match not played?

Zero records a performance in a match played; a match not played provides no performance to count.

Does knowing one class section's mean height determine another section's mean if their sizes match?

No. Equal numbers of students do not provide the actual heights needed to calculate the other section's mean.