Is it correct to say: Using a class limit instead of the class mark as xᵢ?

xᵢ is the mid-value of the class: (lower limit + upper limit)/2.

Is it correct to say: Forgetting to multiply ū by g in the step deviation method?

X̄ = A + g·ū. Writing X̄ = A + ū gives a badly wrong answer.

Is it correct to say: Taking cf of the median class itself in the median formula?

cf is the cumulative frequency of the class just BEFORE the median class.

Is it correct to say: Using the class limits of inclusive classes (like 30–39) directly as L?

Make the classes continuous first; L for 30–39 becomes 29.5.

Is it correct to say: Swapping f₀ and f₂, or writing the denominator as 2f₁ + f₀ + f₂?

f₀ is the class before the modal class, f₂ the class after. The denominator is 2f₁ − f₀ − f₂.

Is it correct to say: Central angles of a pie diagram that do not add up to 360°?

Compute each angle as (component/total) × 360° and check the sum before drawing the sectors.

📄 Formula sheet (PDF)

Statistics

Mean of grouped data (direct, assumed mean and step deviation methods), median, mode, histogram, frequency polygon and pie diagram — Maharashtra SSC Std X Algebra Ch 6

📐 16 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

💡

Board Exam Tips

→Draw a neat table with one column for each quantity (class, xᵢ, fᵢ, dᵢ or uᵢ, fᵢdᵢ or fᵢuᵢ, cf). Most marks are for the table, so keep it tidy.
→Use the method the question asks for. If no method is named, the step deviation method is usually fastest for equal class widths.
→Choose the assumed mean A as a class mark near the middle of the table, preferably one with a large frequency.
→For median and mode the classes must be continuous. Convert classes like 10–19, 20–29 to 9.5–19.5, 19.5–29.5 before reading L.
→For a pie diagram, show the central angle calculation for every component and check that the angles add up to 360°.

📐 Formulas(16)

Class Mark

x_{i} = \frac{Lower class limit + Upper class limit}{2}

Mean: Direct Method

X ˉ = \frac{\sum f _{i} x _{i}}{\sum f _{i}}

Symbol	Meaning
$X ˉ$	Mean of the data
$x_{i}$	Class mark of the i-th class
$f_{i}$	Frequency of the i-th class

Deviation from Assumed Mean

d_{i} = x_{i} - A

Mean: Assumed Mean Method★ Board fav

d ˉ = \frac{\sum f _{i} d _{i}}{\sum f _{i}}, X ˉ = A + d ˉ

Symbol	Meaning
$A$	Assumed mean
$d_{i}$	Deviation of class mark from A
$d ˉ$	Mean of the deviations

Step Deviation

u_{i} = \frac{x _{i} - A}{g}

Mean: Step Deviation Method★ Board fav

u ˉ = \frac{\sum f _{i} u _{i}}{\sum f _{i}}, X ˉ = A + g u ˉ

Symbol	Meaning
$u_{i}$	Step deviation of the i-th class
$g$	G.C.D. of all dᵢ (usually the class width)
$u ˉ$	Mean of the step deviations

Making Classes Continuous

Correction = \frac{( lower limit of next class ) - ( upper limit of a class )}{2}

Cumulative Frequency (less than type)

c f_{k} = f_{1} + f_{2} + \dots + f_{k}

Median Class

Median class = first class with c f \geq \frac{N}{2}

Median of Grouped Data★ Board fav

Median = L + [\frac{\frac{N}{2} - c f}{f}] \times h

Symbol	Meaning
$L$	Lower class limit of the median class
$N$	Total frequency (Σfᵢ)
$c f$	Cumulative frequency of the class preceding the median class
$f$	Frequency of the median class
$h$	Class width

Modal Class

Modal class = class with the highest frequency

Mode of Grouped Data★ Board fav

Mode = L + [\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}] \times h

Symbol	Meaning
$L$	Lower class limit of the modal class
$f_{1}$	Frequency of the modal class
$f_{0}$	Frequency of the class preceding the modal class
$f_{2}$	Frequency of the class succeeding the modal class
$h$	Class width

Frequency Polygon

Plot the points (x_{i}, f_{i})

Pie Diagram: Central Angle★ Board fav

θ = \frac{Value of the component}{Total value} \times 36 0^{\circ}

Pie Diagram: Value from the Angle

Value of the component = \frac{θ}{36 0 ^{\circ}} \times Total value

Check for a Pie Diagram

\sum θ = 36 0^{\circ}

✏️ Solved Examples

1Solved Exampleeasy4 steps

A family's monthly expenditure is: Food Rs 4500, Rent Rs 3000, Education Rs 2400, Others Rs 2100. Find the central angles for a pie diagram.

Find the total expenditure.

4500 + 3000 + 2400 + 2100 = 12000

2Solved Exampleboard4 steps

Marks of 50 students — Classes: 0–10, 10–20, 20–30, 30–40, 40–50; Number of students: 6, 10, 14, 12, 8. Find the mean by the step deviation method.

Class marks are 5, 15, 25, 35, 45. Take A = 25 and g = 10.

u_{i} = \frac{x _{i} - 25}{10} : - 2, - 1, 0, 1, 2

3Solved Exampleboard4 steps

For the same data (Classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 6, 10, 14, 12, 8), find the median.

Cumulative frequencies and N/2.

c f : 6, 16, 30, 42, 50; N = 50, \frac{N}{2} = 25

4Solved ExampleHOTS5 steps

Marks of 40 students — Classes: 10–19, 20–29, 30–39, 40–49, 50–59; Number of students: 5, 9, 15, 7, 4. Find the median and the mode.

The classes are not continuous. Correction = (20 − 19)/2 = 0.5, so the classes become 9.5–19.5, 19.5–29.5, 29.5–39.5, 39.5–49.5, 49.5–59.5.

h = 10

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using a class limit instead of the class mark as xᵢ
✓xᵢ is the mid-value of the class: (lower limit + upper limit)/2.
2
Forgetting to multiply ū by g in the step deviation method
✓X̄ = A + g·ū. Writing X̄ = A + ū gives a badly wrong answer.
3
Taking cf of the median class itself in the median formula
✓cf is the cumulative frequency of the class just BEFORE the median class.
4
Using the class limits of inclusive classes (like 30–39) directly as L
✓Make the classes continuous first; L for 30–39 becomes 29.5.
5
Swapping f₀ and f₂, or writing the denominator as 2f₁ + f₀ + f₂
✓f₀ is the class before the modal class, f₂ the class after. The denominator is 2f₁ − f₀ − f₂.
6
Central angles of a pie diagram that do not add up to 360°
✓Compute each angle as (component/total) × 360° and check the sum before drawing the sectors.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Classes: 5–15, 15–25, 25–35, 35–45; frequencies: 3, 5, 8, 4. Find the mean by the direct method.
Q2
Classes: 100–120, 120–140, 140–160, 160–180, 180–200; frequencies: 4, 7, 10, 6, 3. Find the mean by the assumed mean method, taking A = 150.
Q3
Classes: 0–20, 20–40, 40–60, 60–80, 80–100; frequencies: 8, 12, 20, 6, 4. Find the median.
Q4
For the data in the previous question, find the mode.
Q5
In a pie diagram showing 500 students, the central angle for the science stream is 72°. How many students are in the science stream?
Q6
A component makes up 15% of the total. Find its central angle in a pie diagram.

📝 Notes

Statistics

This chapter is about summarising grouped data with a single value — the mean, median or mode — and then showing data as a histogram, frequency polygon or pie diagram.

Three ways to find the mean

All three methods give the same answer; they differ only in the size of the arithmetic.

Direct method: $\bar{X} = \frac{\sum f_ix_i}{\sum f_i}$ . Good for small class marks.
Assumed mean method: subtract a convenient $A$ from every class mark, so $\bar{X} = A + \bar{d}$ .
Step deviation method: also divide each $d_i$ by a common factor $g$ (usually the class width), so the numbers become small integers and $\bar{X} = A + g\bar{u}$ .

A quick self-check: the mean must lie between the lowest and highest class marks. If it does not, recheck the signs of dᵢ or uᵢ.

Median and mode

Both formulas start with the same four steps: make the classes continuous, find the right class (median class from $\frac{N}{2}$ and the cf column, modal class from the highest frequency), read $L$ and $h$ , then substitute. Most lost marks come from picking the wrong cf (it belongs to the class before the median class) or swapping $f_0$ and $f_2$ .

Pie diagram

A pie diagram shows how a total is divided into parts. Each part gets a sector with central angle $\theta = \frac{\text{part}}{\text{total}} \times 360^\circ$ . Make a table of angles, check that they add to $360^\circ$ , then draw the sectors with a protractor and label each one.

Histogram and frequency polygon

For a histogram, draw adjacent rectangles on the class intervals with heights equal to the frequencies. For a frequency polygon, plot (class mark, frequency) and join the points, closing the figure on the X-axis at both ends.