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Andhra Pradesh (BIEAP)Class 10 Mathematics★ AP Board (BSEAP) SSC Mathematics · ~6 marks · Board: AP Board

Statistics

MEAN (direct, assumed mean, step deviation). MEDIAN and MODE for grouped data. OGIVE (cumulative frequency curve). Empirical relationship between mean, median, mode. AP Board Class 10 Mathematics.

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By the end of this chapter you'll be able to…

1Calculate mean of grouped data using direct method, assumed mean method, and step deviation method
2Find the median class and calculate median using the formula
3Find the modal class and calculate mode using the formula
4Apply the empirical relationship: 3 Median = Mode + 2 Mean
5Draw an Ogive (less-than type) and estimate median graphically

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Why this chapter matters

Statistics in AP SSC tests the three measures of central tendency (mean, median, mode) for grouped data. These are real-world skills used in data analysis. The Ogive (cumulative frequency curve) question is a STANDARD 4-mark graphical question in AP SSC that appears almost every year. The empirical relationship 3 Median = Mode + 2 Mean is a 2-mark trick question. Statistics is formulaic once the table structure is mastered — systematic students score full marks.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Statistics

Mean — Three Methods for Grouped Data

Direct: X̄ = Σfx/Σf
Assumed Mean: X̄ = A + Σfd/Σf (d = x−A)
Step Deviation: X̄ = A + h(Σfu/Σf) (u = (x−A)/h)

Median: L + [(N/2 − CF)/f] × h

L = lower limit of median class. CF = cumulative frequency BEFORE median class.

Mode: L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h

f₁ = freq of modal class. f₀ = before. f₂ = after.

Empirical: 3 Median = Mode + 2 Mean

Ogive: Plot upper limits vs cumulative frequencies. Find median GRAPHICALLY.

Mean of Grouped Data — Three Methods in Detail

1. Direct Method

Formula: X̄ = Σfᵢxᵢ / Σfᵢ

Example 1: Find the mean of the following data:

Class Interval	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	12	7	3

CI	xᵢ (midpoint)	fᵢ	fᵢxᵢ
0-10	5	5	25
10-20	15	8	120
20-30	25	12	300
30-40	35	7	245
40-50	45	3	135
Total		Σf = 35	Σfx = 825

X̄ = 825/35 = 23.57 (approx.).

2. Assumed Mean Method

Formula: X̄ = A + Σfᵢdᵢ / Σfᵢ, where dᵢ = xᵢ − A

'This method is useful when the numbers are LARGE. Choose a middle value as the assumed mean A (usually the midpoint of the class with the highest frequency).'

Example 2: Using the same data, let A = 25 (midpoint of 20-30, the class with highest frequency).

CI	xᵢ	fᵢ	dᵢ = xᵢ−25	fᵢdᵢ
0-10	5	5	−20	−100
10-20	15	8	−10	−80
20-30	25	12	0	0
30-40	35	7	10	70
40-50	45	3	20	60
Total		35		Σfd = −50

X̄ = 25 + (−50/35) = 25 − 1.43 = 23.57 ✓.

3. Step Deviation Method

Formula: X̄ = A + h(Σfᵢuᵢ/Σfᵢ), where uᵢ = (xᵢ−A)/h and h = class width.

'Use this method when the class width is UNIFORM. It simplifies calculations dramatically.'

Example 3: Same data, A = 25, h = 10.

CI	xᵢ	fᵢ	uᵢ = (x−25)/10	fᵢuᵢ
0-10	5	5	−2	−10
10-20	15	8	−1	−8
20-30	25	12	0	0
30-40	35	7	1	7
40-50	45	3	2	6
Total		35		Σfu = −5

X̄ = 25 + 10×(−5/35) = 25 − 1.43 = 23.57 ✓.

Median of Grouped Data

Formula: Median = L + [(N/2 − CF)/f] × h

Where:

L = lower limit of median class
N = total frequency
CF = cumulative frequency of the class BEFORE the median class
f = frequency of the median class
h = class width

Step 1: Find the median class — the class where the cumulative frequency first reaches or exceeds N/2.

Example 4: Find the median for the same data.

CI	fᵢ	Cumulative Frequency
0-10	5	5
10-20	8	13
20-30	12	25
30-40	7	32
40-50	3	35
Total	35

N/2 = 35/2 = 17.5. The CF first exceeding 17.5 is 25, so median class = 20-30. L = 20, CF = 13, f = 12, h = 10. Median = 20 + [(17.5 − 13)/12] × 10 = 20 + (4.5/12)×10 = 20 + 3.75 = 23.75.

Mode of Grouped Data

Formula: Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h

Where:

L = lower limit of modal class (class with highest frequency)
f₁ = frequency of modal class
f₀ = frequency of class BEFORE modal class
f₂ = frequency of class AFTER modal class
h = class width

Example 5: Find the mode for the same data. Highest frequency = 12 → modal class = 20-30. L = 20, f₁ = 12, f₀ = 8, f₂ = 7, h = 10. Mode = 20 + [(12−8)/(24−8−7)] × 10 = 20 + (4/9)×10 = 20 + 4.44 = 24.44.

Empirical Relationship

For a moderately skewed distribution: 3 Median = Mode + 2 Mean

Verification from our example: Mean = 23.57, Median = 23.75, Mode = 24.44. Mode + 2 Mean = 24.44 + 47.14 = 71.58. 3 Median = 71.25. Close (difference due to rounding): 71.58 ≈ 71.25. The relationship holds approximately.

'This formula is used to ESTIMATE one measure when the other two are known. It does NOT give exact values.'

Example 6: If mean = 25 and median = 23, find the approximate mode. 3 × 23 = Mode + 2 × 25 → 69 = Mode + 50 → Mode ≈ 19.

Cumulative Frequency Curve (Ogive)

Less Than Ogive

Plot UPPER CLASS LIMITS on x-axis and CUMULATIVE FREQUENCIES on y-axis. Join the points with a smooth curve.

Example 7: For the given data:

CI	Upper Limit	CF (less than)
0-10	10	5
10-20	20	13
20-30	30	25
30-40	40	32
40-50	50	35

Plot points: (10,5), (20,13), (30,25), (40,32), (50,35). A line drawn from N/2 = 17.5 to the curve and down to the x-axis gives the median ≈ 23.75.

More Than Ogive

Plot LOWER CLASS LIMITS vs 'more than' cumulative frequencies (total − CF). The median is the x-coordinate where the two ogives (less than and more than) INTERSECT.

Common Mistakes

Wrong midpoint: Class mark = (upper limit + lower limit)/2. NOT upper − lower.
Median class identification: The median class is where CF FIRST exceeds N/2, NOT equals N/2.
Using the wrong cumulative frequency in median formula: CF is the cumulative frequency BEFORE the median class, NOT of the median class.
Mode formula: wrong order of f₀ and f₂: f₀ is BEFORE the modal class, f₂ is AFTER. Getting them reversed gives a wrong answer.
Ogive: plotting wrong limits: For 'less than ogive' plot UPPER limits. For 'more than ogive' plot LOWER limits.

AP Exam Focus

Topic	Marks	Frequency
Mean (all three methods)	4	Very Common
Median of grouped data	4	Very Common
Mode of grouped data	3-4	Common
Ogive (less than / more than)	4	Common
Empirical relationship	2-3	Moderate

Self-Test Questions

Find the mean of: 10-20 (f=5), 20-30 (f=8), 30-40 (f=12), 40-50 (f=5). (Answer: midpoints 15,25,35,45. X̄ = (75+200+420+225)/30 = 920/30 = 30.67)
Find the median for: 0-10 (f=4), 10-20 (f=6), 20-30 (f=10), 30-40 (f=8), 40-50 (f=2). (Answer: N=30, N/2=15, median class=20-30, L=20, CF=10, f=10, h=10. Median=20+(5/10)×10=25)
Find the mode: 0-10(f=3), 10-20(f=7), 20-30(f=10), 30-40(f=5), 40-50(f=2). (Answer: modal class=20-30. Mode=20+(10-7)/(20-7-5)×10=20+3/8×10=23.75)
If mode = 36 and mean = 30, find median using empirical formula. (Answer: 3M = 36+60 = 96 → Median = 32)
The mean of 5 numbers is 20. If one number is removed, the mean becomes 18. Find the removed number. (Answer: Sum of 5 = 100. Sum of 4 = 72. Removed = 100−72 = 28)
Draw a less than ogive for the data in Q2 and estimate the median.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Statistics Formulas

MEAN — Direct: X̄ = Σfx/Σf (x = midpoint of class). Assumed Mean: X̄ = A + Σfd/Σf (d = x−A). Step Deviation: X̄ = A + h(Σfu/Σf) (u = (x−A)/h, h = class width). MEDIAN: L + [(N/2 − CF)/f] × h. L = lower boundary of median class. N = total freq. CF = cumulative freq of class BEFORE median class. f = freq of median class. h = class width. MODE: L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h. f₁ = freq of modal class (highest freq). f₀ = freq of class BEFORE modal class. f₂ = freq of class AFTER modal class. L = lower boundary of modal class. h = class width. EMPIRICAL: 3 Median = Mode + 2 Mean.

OGIVE (Less-than type): Plot cumulative frequency (y-axis) against upper class boundary (x-axis). The cumulative frequency starts at 0. Join points with a smooth S-curve. TO FIND MEDIAN from ogive: draw a horizontal line at N/2 on y-axis. Where it meets the ogive, drop a vertical line to x-axis → that value is the MEDIAN. AP Board asks this almost every year — practice drawing the ogive carefully.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using class midpoint instead of lower boundary in median/mode formulas

✓ In the median formula: L = LOWER CLASS BOUNDARY of the median class (not the midpoint). The median class is the class where the cumulative frequency FIRST reaches or exceeds N/2. Example: if N=40, N/2=20. Find the class whose cumulative frequency first reaches 20 — that is the median class. L is its lower boundary. CF is the cumulative frequency of the class BEFORE it (not the median class itself).

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Statistics?

1 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

1 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•MEAN — THREE METHODS: Direct method: X̄ = Σfx/Σf (x = class midpoint). Best when numbers are small. Assumed Mean: X̄ = A + Σfd/Σf (d = x−A, A = assumed mean near the middle of data). Best when midpoints are large. Step Deviation: X̄ = A + h(Σfu/Σf) (u = d/h = (x−A)/h, h = class width). Fastest when class width is consistent.
•FINDING CLASS MIDPOINT: For class 10–20, midpoint x = (10+20)/2 = 15. For 20–30, x = 25. For 0–10, x = 5. Always use the midpoint of the class interval as the x value in mean calculations.
•MEDIAN FORMULA: Median = L + [(N/2 − CF)/f] × h. IDENTIFY MEDIAN CLASS: Find N/2. Scan cumulative frequency column — the first class where CF reaches or passes N/2 is the MEDIAN CLASS. L = lower boundary of median class. CF = cumulative frequency of class BEFORE (not of) the median class. f = frequency of median class. h = class width.
•MODE FORMULA: Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h. IDENTIFY MODAL CLASS: Class with highest frequency = modal class. f₁ = frequency of modal class. f₀ = frequency of class BEFORE modal class. f₂ = frequency of class AFTER modal class. L = lower boundary of modal class. h = class width.
•EMPIRICAL RELATIONSHIP: 3 Median = Mode + 2 Mean. Rearranged forms: Median = (Mode + 2 Mean)/3. Mode = 3 Median − 2 Mean. Mean = (3 Median − Mode)/2. AP SSC tests this as a 2-mark question: 'If mean = 45, mode = 51, find median.'
•OGIVE (CUMULATIVE FREQUENCY CURVE): Plot upper class boundary on x-axis, cumulative frequency on y-axis. Points: (upper boundary, cumulative freq). LESS-THAN OGIVE starts at (lower boundary of first class, 0). Connect points with a smooth S-shaped curve. To find MEDIAN from ogive: draw horizontal line at N/2, where it meets ogive draw vertical line to x-axis — that x-value is the median.
•CUMULATIVE FREQUENCY TABLE: Running total of frequencies. Class 0–10 has freq 5, CF = 5. Class 10–20 has freq 8, CF = 5+8 = 13. Class 20–30 has freq 15, CF = 13+15 = 28. The LAST cumulative frequency must equal N (total frequency). This self-check prevents CF table errors.
•WHICH MEAN METHOD TO USE: Large class midpoints (e.g., 150, 160, 170) → use step deviation (fastest). Medium midpoints → assumed mean. Small midpoints (single digits) → direct method. In AP SSC, the assumed mean method is the most frequently required explicitly.
•MODAL CLASS vs MEDIAN CLASS: They are often different classes. Modal class = highest frequency class (no calculation needed — just look). Median class = class where N/2 cumulative frequency first lands. Never confuse them — they require different formulas (mode formula uses f₀, f₁, f₂; median uses CF, f, L of their respective classes).
•GRAPHICAL MEDIAN from TWO OGIVES: Drawing both less-than and more-than ogives on the same graph, their intersection point's x-coordinate gives the median directly. This two-ogive method is tested occasionally — the intersection x-value is always N/2 on the y-axis.

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

AP government welfare schemes and income data

The AP government uses median household income (not mean) for welfare eligibility because median is not distorted by extreme values. If 99 families earn ₹20,000/month and 1 family earns ₹2,000,000/month, the mean income is far higher than what most families actually earn. The median gives a more honest picture of the 'typical' family income. This is why poverty line statistics, BPL (Below Poverty Line) classifications, and scheme eligibility in AP use median-based thresholds.

School exam analysis and student performance tracking

When a teacher analyses class marks, they compute mean (overall performance), median (typical student performance), and mode (most common score). If mean = 65 but mode = 45, many students scored around 45 while a few high scorers pulled the average up — the teacher needs to address the low-scoring group. In AP's Samagra Shiksha programme, district-level performance is analysed using these exact statistics. The modal class in a marks frequency table shows where most students score — directly informing remedial teaching priorities.

Epidemiology and disease surveillance

During disease outbreaks (like AP's annual dengue surveillance or COVID tracking), health officials plot cumulative case counts — exactly an ogive. The ogive's slope at any point shows the outbreak's growth rate. The inflection point (steepest slope) marks the peak of the outbreak. Finding the median incubation period (using the median formula on grouped interval data) helps design quarantine protocols. Statistics from Class 10 is the mathematical tool used in public health surveillance.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Table structure (4 marks): always make a proper table. For mean: columns = Class, Midpoint(x), Frequency(f), d=x−A, fd (for assumed mean) or u=d/h, fu (for step deviation). Total row at bottom. Final formula at end. Each column earns a mark.
Median: identify median class EXPLICITLY — write 'N = [total]. N/2 = [value]. Cumulative frequencies: [list]. Median class = [class range] (first CF ≥ N/2).' Then write the formula, substitute, calculate. Show each substitution step.
Mode: identify modal class EXPLICITLY — write 'Modal class = [range] (highest frequency = f₁ = [value]).' Then identify f₀ and f₂ (classes immediately before and after). Write formula, substitute, calculate.
Ogive (4 marks): plot on graph paper (or neatly on plain paper). Table of (upper boundary, CF). Mark all points accurately. Connect with smooth S-curve. Mark N/2 on y-axis with a dotted horizontal line. Where it meets the ogive, draw a dotted vertical line to x-axis. Circle the median value. These markings show method and earn marks even if the curve has minor errors.
Empirical formula (2 marks): just substitute and calculate. Write the formula, substitute the given values, compute. State 'Median ≈ [value]' — the approximately equal (≈) is correct since empirical formulas are approximate.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research the difference between descriptive and inferential statistics — descriptive statistics (mean, median, mode, range, variance) summarise the DATA YOU HAVE. Inferential statistics allows you to make conclusions about a POPULATION from a SAMPLE (using probability and sampling theory). For example, the National Sample Survey Organisation (NSSO) interviews ~100,000 households to infer income levels of India's 300 million households — using inferential statistics. Research how sample size affects the reliability of inferences (Central Limit Theorem).
Investigate the concept of variance and standard deviation — mean tells you the centre, but how spread out is the data? Variance = Σf(x−X̄)²/N, standard deviation σ = √Variance. For the same mean, two datasets can be very differently spread. Research the 68-95-99.7 rule (empirical rule) for normal distributions: 68% of data lies within 1 standard deviation of the mean, 95% within 2, 99.7% within 3. This is why in cricket, a batsman whose average is 50 with std deviation 5 is MORE reliable than one with average 50 and std deviation 30.
Explore the Lorenz Curve and Gini Coefficient — a graphical tool (similar to an ogive) to measure income inequality in a country. The x-axis shows cumulative share of population (from poorest), y-axis shows cumulative share of income. A perfectly equal society gives a straight diagonal line; actual curves bow below it. The Gini coefficient = area between the diagonal and the actual curve ÷ area of triangle below diagonal. India's Gini coefficient is ~0.35 (moderate inequality). Research AP's specific inequality data.
Research big data and how AI systems use statistical measures — modern recommendation systems (Netflix, YouTube, Spotify) compute mean engagement times, median session lengths, and modal content categories for millions of users to personalise recommendations. Machine learning models find the 'mean' of high-dimensional data (centroids in k-means clustering). Research how the statistical concepts from Class 10 — mean, median, mode, frequency distributions — are the foundational building blocks of data science and AI.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC (Class 10)Very High — Statistics (mean by assumed mean, median, mode, ogive) is a standard 4–8 mark section in every AP SSC Mathematics paper

JEE Main (Statistics)Medium — Statistics (mean, variance, standard deviation of grouped data) appears in JEE; Class 10 provides the data handling foundation

NTSE (Mathematics)Medium — Statistics problems appear in NTSE Stage I

AP EAPCET and Commerce streamHigh — Statistics (measures of central tendency, dispersion) is a major chapter in both EAPCET Mathematics and Class 11–12 Commerce statistics

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Questions students ask

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STEP 1: Sum all frequencies to get N. STEP 2: Compute N/2. STEP 3: Build the cumulative frequency column. STEP 4: Scan down the CF column — find the FIRST class where the CF REACHES OR EXCEEDS N/2. That entire class is the MEDIAN CLASS. IMPORTANT: It's the first class that reaches N/2, not the class that straddles N/2 in some way. If N=40, N/2=20. If CF values are 5, 13, 28, 38, 40 — the class with CF=28 is the first to reach or exceed 20, so THAT class is the median class. CF used in the formula is the one BEFORE this class: CF=13 (not 28).

RAW DATA MODE: Simply the value that appears most frequently. In {1, 2, 2, 3, 4, 2, 5}, mode = 2 (appears 3 times). No formula needed. GROUPED DATA MODE: You cannot find the exact mode because individual values are grouped into classes. Instead, you find the MODAL CLASS (highest frequency class) and use the formula: Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h. This gives an estimated mode within the modal class. The formula interpolates based on how the modal class frequency compares to its neighbouring classes — a higher frequency class on one side 'pulls' the mode toward that side.

The OGIVE (cumulative frequency curve) is S-shaped because: (1) At first, cumulative frequency grows slowly (few data points in early classes). (2) In the MIDDLE, it grows fastest (most data is near the mean, in the central classes — the steep part of the S). (3) At the end, it slows again (few data points in extreme high classes). This S-shape is characteristic of approximately NORMAL data. If data is skewed, the S is asymmetric — steeper at the low end (right-skewed) or steep at the high end (left-skewed). The median (N/2 on y-axis) always corresponds to the inflection point of the ogive — the middle of the S.

No — the answer is the same regardless of what A you choose. The assumed mean method is EXACT: X̄ = A + Σfd/Σf gives the correct mean for any A. However, CHOOSING A WISELY makes the arithmetic easier: (1) Choose A close to the middle of the data (near the expected mean). (2) Ideally choose one of the class midpoints. (3) A convenient choice minimises the size of d values (d = x − A). If d values are large, the Σfd calculation involves bigger numbers and more chance of arithmetic error. For AP SSC: typically choose the midpoint of the class with highest frequency (the modal class midpoint) as A.

Empirical formula: 3 Median = Mode + 2 Mean = 15 + 2(20) = 15 + 40 = 55. Median = 55/3 ≈ 18.33. Alternatively: Median = (Mode + 2 Mean)/3 = (15 + 40)/3 = 55/3 ≈ 18.33. Check reasonableness: median (18.33) should be between mode (15) and mean (20) — it is ✓. The empirical formula gives an APPROXIMATE relationship that holds for moderately skewed data.

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