Statistics ICSE Class 9: Data, Mean, Median and Frequency Polygons

What Statistics is about

Weightage: CISCE publishes no chapter weightage and no question-paper layout for ICSE Class 9 Mathematics. Class IX is examined by your school: one 80-mark paper of 3 hours plus 20 marks of internal assessment. Statistics is in the 5. Statistics part of the 2028 syllabus.Source: CISCE ICSE 2028 Mathematics syllabus, Class IX

Key concepts

Raw, arrayed and grouped data; tally marks

• Raw data: observations in the order they were collected.
• Arrayed data: the same observations arranged in ascending or descending order.
• Grouped data: observations sorted into classes (such as 0–10, 10–20) with the number in each class, its frequency.

To tabulate raw data, go through the list once and put a tally mark | against the value or class for each observation. Every fifth mark is drawn across the previous four, making bundles of five that are easy to count.

Worked example

The numbers of goals scored by a team in 15 matches are: 2, 0, 1, 3, 2, 1, 1, 4, 2, 0, 1, 2, 3, 1, 2. Make a frequency table with tally marks.

Try it first, then show the solutionSolution

1. Values 0 to 4. Tally each observation once.
2. 0: || (2); 1: |||| (5, drawn as a bundle); 2: |||| (5); 3: || (2); 4: | (1).
3. Check: $2+5+5+2+1=15$ ✓.

Answer: Frequencies: 0→2, 1→5, 2→5, 3→2, 4→1

Discrete and continuous variables

A variable is a quantity that changes from one observation to another.

• A discrete variable takes only separate values, usually whole numbers obtained by counting: number of children in a family, goals in a match.
• A continuous variable can take any value in a range, obtained by measuring: height, mass, temperature, time.

Continuous data is grouped in classes with no gaps (10–20, 20–30, …).

Worked example

Classify as discrete or continuous: (i) number of pages in books (ii) heights of students (iii) marks out of 20 (iv) time taken to run 100 m.

Try it first, then show the solutionSolution

1. Counted: (i) and (iii) are discrete.
2. Measured: (ii) and (iv) are continuous.

Answer: (i) discrete (ii) continuous (iii) discrete (iv) continuous

Class intervals: limits, boundaries, size and mark

For a class such as 20–29:

• lower limit 20 and upper limit 29;
• class mark (mid-value) $=\frac{\text{lower limit}+\text{upper limit}}{2}=24.5$;
• class size (width) $=$ difference between the lower limits of consecutive classes $=10$.

In exclusive (continuous) form, such as 20–30, 30–40, the upper limit is not included in the class: 30 goes into 30–40.

In inclusive (discontinuous) form, such as 20–29, 30–39, there are gaps. Class boundaries close the gaps: half the gap is subtracted from each lower limit and added to each upper limit. Here the gap is $30-29=1$, so the adjustment is $0.5$: 19.5–29.5, 29.5–39.5.

Worked example

Convert the classes 10–19, 20–29, 30–39 into continuous form and give the class mark and size of the first class.

Try it first, then show the solutionSolution

1. Gap $=20-19=1$; adjustment $=0.5$.
2. Continuous classes: 9.5–19.5, 19.5–29.5, 29.5–39.5.
3. Class mark $=\frac{10+19}{2}=14.5$ (unchanged); class size $=10$.

Answer: 9.5–19.5, 19.5–29.5, 29.5–39.5; mark 14.5; size 10

Mean of ungrouped data

Raw data: mean $\bar{x}=\frac{\text{sum of observations}}{\text{number of observations}}=\frac{\sum x}{n}$.

Frequency table (each value $x$ with frequency $f$): $\bar{x}=\frac{\sum fx}{\sum f}$.

Make three columns: $x$, $f$, $fx$, and total the $f$ and $fx$ columns. Useful facts:

• If every observation is increased by $k$, the mean increases by $k$.
• Sum of observations $=n\times \bar{x}$, which helps with missing or wrongly copied values.

Worked example

The mean of 25 observations is 36. Later it is found that 43 was copied wrongly as 34. Find the correct mean.

Try it first, then show the solutionSolution

1. Wrong sum $=25\times 36=900$.
2. Correct sum $=900-34+43=909$.
3. Correct mean $=\frac{909}{25}=36.36$.

Answer: 36.36

Median of ungrouped data

The median is the middle value when the data is arranged in order.

For $n$ observations in order:

• $n$ odd: median $=$ the $\left(\frac{n+1}{2}\right)$th value;
• $n$ even: median $=$ the mean of the $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th values.

For a frequency table, add a cumulative frequency (cf) column; the median is the value whose cf first reaches or passes the required position.

The median is not affected by one very large or very small value, which is why it is used for data such as incomes.

Worked example

Find the median of 12, 15, 9, 20, 18, 11.

Try it first, then show the solutionSolution

1. Arrange: 9, 11, 12, 15, 18, 20 ($n=6$, even).
2. 3rd and 4th values: 12 and 15.
3. Median $=\frac{12+15}{2}=13.5$.

Answer: 13.5

Frequency polygon

A frequency polygon is a line graph of a frequency distribution.

1. Make the classes continuous if they are not.
2. Find each class mark.
3. Plot points (class mark, frequency).
4. Add an imaginary class of frequency 0 at each end, and plot its class mark on the $x$-axis.
5. Join the points with straight line segments.

It can also be drawn by joining the mid-points of the tops of the bars of a histogram. Choose a sensible scale and write it on the graph; if the first class does not start near 0, use a kink (zig-zag) on the $x$-axis.

Worked example

Write down the points to plot for a frequency polygon of: 10–20: 4, 20–30: 9, 30–40: 6.

Try it first, then show the solutionSolution

1. Class marks: 15, 25, 35.
2. Points: (15, 4), (25, 9), (35, 6).
3. Imaginary classes 0–10 and 40–50 with frequency 0: points (5, 0) and (45, 0). Join in order.

Answer: (5, 0), (15, 4), (25, 9), (35, 6), (45, 0)

Maths in real life: Statistics

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Grouped Data, Class Intervals, Frequency Tables and the Frequency Polygon
• Mean and Median of Ungrouped Data

Statistics formula sheet

Mean of raw data

$\bar{x}=\frac{\sum x}{n}$

Sum ÷ number.

Mean of a frequency table

$\bar{x}=\frac{\sum fx}{\sum f}$

Ungrouped values x with frequencies f.

Median position (n odd)

$\left(\frac{n+1}{2}\right)\text{th value}$

Data in order.

Median (n even)

$\frac{1}{2}\left[\frac{n}{2}\text{th}+\left(\frac{n}{2}+1\right)\text{th}\right]$

Average of the two middle values.

Class mark

$\frac{\text{lower limit}+\text{upper limit}}{2}$

Plotting point for the polygon.

Boundary adjustment

$\frac{1}{2}(\text{next lower limit}-\text{this upper limit})$

Subtract from lower, add to upper limits.

Statistics: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Statistics

The question patterns to expect, and the method that works for each.

Tricks to remember Statistics

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Count = discrete, measure = continuous

  If you count it on fingers, it is discrete; if you need a ruler, scale or clock, it is continuous.

  Use it when: Classifying variables.

• Sort before the middle

  No median without arranging the data first.

  Use it when: Median of raw data.

• Half the gap

  Boundaries: take half the gap off each lower limit and add it to each upper limit.

  Use it when: Inclusive to continuous classes.

• Polygons touch the floor

  A frequency polygon starts and ends on the x-axis at the empty neighbouring classes.

  Use it when: Drawing the polygon.

Statistics: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Statistics

The slips examiners see most often, and what to write instead.

• ✗Mistake: Finding the median without arranging the data

  The middle of the list as written is not the median unless the list is in order.

  Wrong working

  3, 8, 5, 12, 9, 6, 10: median $=12$

  Right working

  3, 5, 6, 8, 9, 10, 12: median $=8$

  ✓Do this: Always write the arranged list first.

  Usually costs: The whole answer.

• ✗Mistake: Median from a frequency table without cumulative frequency

  CISCE examiners' comments, ICSE Mathematics 2025 (Class X): for the median of a frequency table, some candidates averaged the frequencies, some averaged the median ranks, and others gave no cumulative frequency column.

  ✓Do this: Add a cf column; find the position ($\frac{n+1}{2}$ or the two middle positions); read the value where cf first reaches it.

  Usually costs: Most of the median marks.

• ✗Mistake: Errors in the fx column and rounding

  CISCE examiners' comments, ICSE Mathematics 2025 (Class X): some candidates made calculation errors in $fx$ and did not round the mean as asked.

  ✓Do this: Check each $fx$ product and the totals; round only at the end, to the accuracy the question asks.

  Usually costs: 1 to 2 marks.

• ✗Mistake: Dividing by the number of values instead of Σf

  In a frequency table there are $\sum f$ observations, not as many as there are rows.

  ✓Do this: Divide $\sum fx$ by $\sum f$.

  Usually costs: The answer.

• ✗Mistake: Frequency polygon not closed or freehand

  Missing end points on the x-axis or curved joins lose graph marks.

  ✓Do this: Add the empty classes at both ends and join with a ruler.

  Usually costs: 1 mark.

Practice questions on Statistics

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which of these is a continuous variable?
   (a) number of goals
   (b) shoe size
   (c) mass of a parcel
   (d) number of siblings

   Hint

   Measured, not counted.

   Answer

   (c)

2. Q2 · 1 mark · Short answerExam level

   Find the mean of the first ten natural numbers.

   Hint

   Sum 55.

   Answer

   5.5

3. Q3 · 2 marks · Short answerExam level

   Find the median of 25, 16, 26, 16, 32, 31, 19, 28, 35.

   Hint

   Arrange; n = 9.

   Answer

   16, 16, 19, 25, 26, 28, 31, 32, 35: median 26.

4. Q4 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): The class mark of 10–19 in an inclusive table is the same as that of 9.5–19.5.
   Reason (R): Adjusting class limits to boundaries moves both ends by the same amount in opposite directions.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Compute both class marks.

   Answer

   (c): both are 14.5, because subtracting 0.5 from one end and adding 0.5 to the other leaves the mid-value unchanged.

5. Q5 · 2 marks · Short answerHarder exam level

   The mean of 10 observations is 20. If each is multiplied by 3 and then 2 is subtracted, find the new mean.

   Hint

   The mean changes in the same way.

   Answer

   $3\times 20-2=58$.

6. Q6 · 4 marks · Long answerHarder exam level

   Find the mean and median of: $x$: 2, 3, 4, 5, 6, 7 with $f$: 3, 5, 9, 6, 4, 3.

   Hint

   Σf = 30; median: mean of the 15th and 16th values.

   Answer

   $\sum fx=6+15+36+30+24+21=132$; mean $=4.4$. cf: 3, 8, 17, 23, 27, 30: 15th and 16th values are both 4, so median $=4$.

7. Q7 · 3 marks · Short answerHarder exam level

   The mean of 6, 8, $x+2$, 10, $2x-1$ and 2 is 9. Find $x$ and the median of the data.

   Hint

   Sum = 54.

   Answer

   $27+3x=54$, $x=9$. Data: 6, 8, 11, 10, 17, 2 → 2, 6, 8, 10, 11, 17; median $=9$.

8. Q8 · 2 marks · Short answerExam level

   Convert into continuous classes and find the class size: 50–54, 55–59, 60–64, 65–69.

   Hint

   Gap 1.

   Answer

   49.5–54.5, 54.5–59.5, 59.5–64.5, 64.5–69.5; class size 5.

9. Q9 · 3 marks · Short answerHarder exam level

   The median of the ordered data 11, 13, 15, 19, $x+2$, $x+4$, 30, 35, 39, 46 is 24. Find $x$.

   Hint

   Mean of the 5th and 6th values.

   Answer

   $\frac{(x+2)+(x+4)}{2}=24\Rightarrow x+3=24\Rightarrow x=21$.

10. Q10 · 4 marks · Long answerHarder exam level

    Heights (cm) of 30 plants: 10–20: 3, 20–30: 7, 30–40: 10, 40–50: 6, 50–60: 4. List the points you would plot for a frequency polygon, and state how many plants are at least 30 cm tall.

    Hint

    Class marks 15, 25, …; add end points.

    Answer

    (5, 0), (15, 3), (25, 7), (35, 10), (45, 6), (55, 4), (65, 0). At least 30 cm: $10+6+4=20$ plants.

11. Q11 · 3 marks · Short answerHarder exam level

    The mean of 20 observations is 15. On checking, it was found that 3 had been taken as 8 and 15 as 25; the other observations are correct. Find the correct mean.

    Hint

    Correct the total.

    Answer

    Wrong total $=300$. Correct total $=300-8-25+3+15=285$. Mean $=14.25$.

12. Q12 · 4 marks · Long answerHOTS / competency

    The mean of five numbers is 18, and their median is 20. If the smallest is removed, the mean becomes 21. Find the smallest number, and explain whether the median of the remaining four must be at least 20.

    Hint

    Totals 90 and 84.

    Answer

    Total $=90$; after removal $4\times 21=84$, so the smallest is 6. In order the five are $6,b,20,d,e$ with $b\le 20\le d$. The remaining four, $b,20,d,e$, have median $\frac{20+d}{2}\ge 20$, since $d\ge 20$. So yes.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is arrayed data?Show answer

   Raw data arranged in ascending or descending order.

2. 2.Is the number of students in a class discrete or continuous?Show answer

   Discrete.

3. 3.Class mark of 15–25?Show answer

   20.

4. 4.Formula for the mean of a frequency table?Show answer

   Σfx ÷ Σf.

5. 5.Median of 2, 4, 6, 8?Show answer

   5.

6. 6.Boundaries of the class 20–29 when the next class is 30–39?Show answer

   19.5 and 29.5.

7. 7.What is plotted for a frequency polygon?Show answer

   (class mark, frequency), closed at both ends on the x-axis.

8. 8.Mean of 5, 8, 12, 15, 20?Show answer

   12.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you tell raw, arrayed and grouped data apart?
• ☐Can you make a tally table and check the total?
• ☐Can you convert inclusive classes to continuous form?
• ☐Do you use Σfx ÷ Σf for a frequency table?
• ☐Do you arrange data before finding the median?
• ☐Do you add a cumulative frequency column for the median?
• ☐Do you close the frequency polygon on the x-axis?

Statistics: frequently asked questions