What is Statistics for Business in A/L Examination (Sri Lanka)?

Statistics for Business is a topic in the Commerce Stream syllabus of A/L Examination (Sri Lanka). It carries a weight of 3% in the exam. Our notes cover the key concepts, formulas, and problem-solving approaches you need to master this topic.

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Statistics for Business

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Statistics for Business — Key Facts for Sri Lanka A/L Examination

Measures of Central Tendency:

Measure	Formula	Best Use
Mean	Sum of values / n	Symmetric data, no outliers
Median	Middle value (after sorting)	Skewed data, ordinal data
Mode	Most frequent value	Categorical data, peak value

Dispersion Measures:

Measure	Formula	What it measures
Range	Max - Min	Simplest, ignores distribution
Variance	Σ(x-x̄)²/(n-1)	Average squared deviation
Standard Deviation	√Variance	Same unit as data
Coefficient of Variation	(SD/Mean)×100	Relative variability

⚡ A/L Exam Tip: When a Sri Lankan business question mentions “average,” always ask: mean, median, or mode? They can give very different answers with skewed data!

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Statistics for Business — Detailed Study Guide

Data Classification and Presentation

Types of Data:

Classification	Type	Example
By source	Primary	Survey results from Sri Lankan consumers
By source	Secondary	Census data, past exam results
By nature	Quantitative	Monthly sales in Rs.
By nature	Qualitative	Product type (textile, food)
By time	Time series	Tea export prices 2020-2024
By time	Cross-sectional	Sales of different shops on one day

Classifying Raw Data:

Raw Data: Randomly recorded observations
Frequency Distribution: Summary table showing frequencies

Steps:
1. Find range = Maximum - Minimum
2. Decide number of classes (usually 5-15)
3. Class width = Range / Number of classes
4. Set class boundaries (avoid gaps)
5. Tally and count frequencies

Example - Sri Lankan Business Scenario:

Monthly sales (in Rs. '000) for 40 small businesses:
45, 52, 38, 61, 55, 47, 63, 58, 42, 49
55, 67, 72, 48, 53, 60, 44, 56, 65, 51
58, 63, 71, 49, 54, 46, 62, 57, 43, 50
59, 64, 69, 53, 47, 56, 61, 54, 48, 60

Range = 72 - 38 = 34
Classes (6 classes): 36-41, 42-47, 48-53, 54-59, 60-65, 66-71, 72-77
Width = 34/6 ≈ 6 (use width 6)

Class     | Tally           | Frequency
----------|-----------------|----------
36 - 41   | I               | 1
42 - 47   | IIII I          | 6
48 - 53   | IIII IIII       | 9
54 - 59   | IIII IIII I     | 11
60 - 65   | IIII IIII       | 9
66 - 71   | II              | 3
72 - 77   | I               | 1
          |                 | ---
          |                 | 40

Graphical Presentation:

Chart	Use	Sri Lankan Example
Histogram	Continuous data frequency	Distribution of tea export weights
Frequency Polygon	Comparing distributions	Monthly sales comparison
Bar Chart	Categorical data	Exports by province
Pie Chart	Parts of a whole	Sri Lanka export composition
Ogive	Cumulative frequency	Below/above threshold analysis
Line Chart	Time series	Colombo Stock Exchange index over 5 years

Measures of Central Tendency

Mean (Arithmetic Average):

Ungrouped: x̄ = Σx / n
Grouped: x̄ = Σ(f×x) / Σf

Where x = class midpoint, f = frequency

Example (Ungrouped):

Monthly salary (Rs. '000) of 10 employees:
40, 45, 50, 55, 60, 65, 70, 80, 120, 200
Mean = (40+45+50+55+60+65+70+80+120+200) / 10 = 785/10 = Rs. 78,500

Example (Grouped):

Class (Rs.'000) | Midpoint (x) | Freq (f) | f×x
----------------|--------------|----------|------
30 - 39         | 34.5         | 5        | 172.5
40 - 49         | 44.5         | 12       | 534.0
50 - 59         | 54.5         | 18       | 981.0
60 - 69         | 64.5         | 10       | 645.0
70 - 79         | 74.5         | 5        | 372.5
                |              | Σf = 50 | Σfx = 2,705

Mean = 2,705 / 50 = 54.1 (Rs. '000) = Rs. 54,100

Median (Middle Value):

Ungrouped: Arrange data, find middle position
If n is odd: Median = value at position (n+1)/2
If n is even: Median = average of values at n/2 and n/2+1

Grouped: Median = L + [(n/2 - cummf) / f] × w

Where:
L = Lower boundary of median class
cummf = Cumulative frequency before median class
f = Frequency of median class
w = Class width

Example (Grouped):

Same distribution, find median:
Σf = 50, so n/2 = 25th value
Cumulative frequencies: 5, 17, 35, 45, 50
Median class = 50 - 59 (cumulative 17 to 35)

L = 49.5, cummf = 17, f = 18, w = 10
Median = 49.5 + [(25 - 17) / 18] × 10
        = 49.5 + (8/18) × 10
        = 49.5 + 4.44 = Rs. 53,940

Mode:

Ungrouped: Most frequently occurring value

Grouped: Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × w

Where:
L = Lower boundary of modal class
f1 = Frequency of modal class
f0 = Frequency before modal class
f2 = Frequency after modal class
w = Class width

⚡ A/L Key: The mode formula is complex — for A/L, students can identify the modal class (highest frequency) and estimate from there. Exact calculation often not required in basic questions.

Relationship Between Mean, Median, Mode:

Distribution	Relationship	Sri Lankan Example
Symmetric/Normal	Mean ≈ Median ≈ Mode	Tea leaf weights
Positively skewed (right)	Mode < Median < Mean	Income distribution
Negatively skewed (left)	Mean < Median < Mode	Failure rates

Measures of Dispersion

Range:

Range = Maximum Value - Minimum Value

Variance and Standard Deviation:

Population Variance: σ² = Σ(x-μ)² / N
Sample Variance: s² = Σ(x-x̄)² / (n-1)
Standard Deviation: σ or s = √Variance

Shortcut formula for ungrouped data:
s² = [Σx² - (Σx)²/n] / (n-1)

Example:

Sales (Rs. '000) for 5 shops: 50, 55, 60, 65, 70
Mean = 300/5 = 60

Deviations: -10, -5, 0, +5, +10
Squared deviations: 100, 25, 0, 25, 100
Sum = 250

s² = 250 / (5-1) = 62.5
s = √62.5 = 7.91 (Rs. '000)

Interpretation: On average, sales deviate from mean by Rs. 7,910

Coefficient of Variation (CV):

CV = (Standard Deviation / Mean) × 100

Used to compare variability of different datasets
Lower CV = More consistent/stable
Higher CV = More variable/risky

Example:

Shop A: Mean = Rs. 500,000, SD = Rs. 50,000 → CV = 10%
Shop B: Mean = Rs. 200,000, SD = Rs. 40,000 → CV = 20%
Shop A is more consistent despite larger absolute variation

Probability

Basic Probability Rules:

P(A) = Favourable outcomes / Total outcomes
P(A) = 0 to 1 (0 = impossible, 1 = certain)

Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
Multiplication Rule: P(A and B) = P(A) × P(B) (independent events)

Example - Sri Lankan Business Application:

A garment factory has two production lines:
- Line A produces 60% of output, 2% defective
- Line B produces 40% of output, 5% defective

P(Defective) = P(A)×P(Defective|A) + P(B)×P(Defective|B)
             = (0.60 × 0.02) + (0.40 × 0.05)
             = 0.012 + 0.020 = 0.032 = 3.2%

Expected Value:

Expected Value = Σ(P × Outcome)
E(X) = Σ(xi × Pi)

Used for decision-making under uncertainty

Example:

A Sri Lankan investor is evaluating a business venture:
Outcome      | Probability | Profit (Rs.)
-------------|-------------|--------------
High demand  | 0.25        | 500,000
Medium demand| 0.50        | 150,000
Low demand   | 0.25        | -100,000

EV = (0.25 × 500,000) + (0.50 × 150,000) + (0.25 × -100,000)
   = 125,000 + 75,000 - 25,000 = Rs. 175,000

Decision: Positive EV suggests the venture is profitable on average

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Statistics for Business — Complete Notes for A/L Sri Lanka

Time Series Analysis

Components of Time Series:

Component	Description	Sri Lankan Example
Trend (T)	Long-term direction	Gradual increase in Sri Lanka tea exports
Seasonal (S)	Regular patterns within a year	Higher clothing sales Dec-Jan
Cyclical (C)	Business cycle fluctuations	GDP growth cycles 5-7 years
Irregular/Random (I)	Unpredictable variations	Effect of 2019 Easter attacks on tourism

Additive Model: Y = T + S + C + I Multiplicative Model: Y = T × S × C × I

Moving Averages (for trend extraction):

3-Period Moving Average:
Period 1-3 Average → centres at period 2
Period 2-4 Average → centres at period 3
And so on...

Centered Moving Average (for even periods):
For 4-period MA, must center by averaging two successive MAs

Example:

Year | Sales (Rs.M) | 3-Period MA
-----|--------------|-------------
2020 | 45           | -
2021 | 52           | (45+52+58)/3 = 51.7
2022 | 58           | (52+58+63)/3 = 57.7
2023 | 63           | (58+63+55)/3 = 58.7
2024 | 55           | (63+55+70)/3 = 62.7
2025 | 70           | (55+70+75)/3 = 66.7
2026 | 75           | -

Seasonal Variation (using multiplicative model):

Step 1: Calculate trend (moving average)
Step 2: Calculate Ratio-to-Moving-Average = Actual / Trend
Step 3: Average the ratios for each season (removes irregular)
Step 4: Adjust so seasonal indices average to 1 (or 100)

Seasonal Index > 1: Above trend in that season
Seasonal Index < 1: Below trend in that season

Example - Quarterly Tea Exports:

Quarter | 2023 Ratio | 2024 Ratio | 2025 Ratio | Average | Seasonal Index
--------|------------|------------|------------|---------|--------------
Q1      | 0.85       | 0.88       | 0.90       | 0.877   | 0.88
Q2      | 0.95       | 0.97       | 0.94       | 0.953   | 0.95
Q3      | 1.05       | 1.08       | 1.06       | 1.063   | 1.07
Q4      | 1.15       | 1.12       | 1.18       | 1.150   | 1.15

Interpretation: Q4 shows highest tea export demand (harvest season)
Q1 shows lowest (off-peak period)

Correlation and Regression

Scatter Diagram:

Pattern	Correlation	Sri Lankan Example
Upward sloping	Positive	Advertising spend vs. sales
Downward sloping	Negative	Debt ratio vs. profitability
No pattern	Zero	Unrelated variables
Perfect line	Perfect correlation	Rare in real data

Karl Pearson’s Coefficient of Correlation (r):

r = Σ(x-x̄)(y-ȳ) / √[Σ(x-x̄)² × Σ(y-ȳ)²]
r = [nΣxy - ΣxΣy] / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]

Interpretation:

r value	Interpretation
+1.0	Perfect positive correlation
+0.7 to +0.99	Strong positive
+0.4 to +0.69	Moderate positive
+0.1 to +0.39	Weak positive
0	No correlation
-0.1 to -0.39	Weak negative
-0.4 to -0.69	Moderate negative
-0.7 to -0.99	Strong negative
-1.0	Perfect negative correlation

Example - Sri Lankan Business:

Advertising (Rs.M) vs. Sales (Rs.M):
x = Advertising: 10, 15, 20, 25, 30
y = Sales: 50, 65, 70, 85, 100

n = 5
Σx = 100, Σy = 370, Σxy = 8,000, Σx² = 2,250, Σy² = 29,450

r = [5(8000) - 100(370)] / √[(5×2250 - 10000)(5×29450 - 136900)]
  = [40000 - 37000] / √[(11250-10000)(147250-136900)]
  = 3000 / √[1250 × 10350]
  = 3000 / √12937500
  = 3000 / 3596.87 = 0.834 (Strong positive correlation)

Linear Regression:

Regression Line: y = a + bx

b = [nΣxy - ΣxΣy] / [nΣx² - (Σx)²]
a = ȳ - b × x̄

Where b = slope (change in y for each unit change in x)
a = intercept (y value when x = 0)

Example:

Using same data:
b = [5(8000) - 100(370)] / [5(2250) - 10000]
  = 3000 / 1250 = 2.4

a = 370/5 - 2.4 × 100/5 = 74 - 2.4 × 20 = 74 - 48 = 26

Regression equation: y = 26 + 2.4x

Interpretation: For every Rs. 1 million increase in advertising,
sales increase by Rs. 2.4 million. With no advertising, base sales = Rs. 26M.

⚡ A/L Exam Tip: In A/L questions, regression is often tested with prediction: “If advertising is Rs. 35 million, predict sales.” Sub x=35 into y = a + bx.

Index Numbers

Simple Index:

Price Index (Simple) = (Pn / P0) × 100
Quantity Index (Simple) = (Qn / Q0) × 100

Weighted Index Numbers:

Laspeyres Index (uses base year quantities — common):

PL = Σ(Pn × Q0) / Σ(P0 × Q0) × 100

Advantages: Uses fixed weights, easier to compute
Disadvantages: May overstate price changes (doesn't reflect substitution)

Paasche Index (uses current year quantities):

PP = Σ(Pn × Qn) / Σ(P0 × Qn) × 100

Advantages: Reflects current consumption patterns
Disadvantages: Weights change each period, harder to compare

Fisher’s Ideal Index (geometric mean of Laspeyres and Paasche):

Fisher = √(Laspeyres × Paasche)

Satisfies factor reversal test and time reversal test
Theoretically ideal but rarely used in practice

Example - Sri Lanka Tea Export Prices:

Product  | Base Year         | Current Year
         | Price (Rs./kg) Qty (M kg) | Price (Rs./kg) Qty (M kg)
----------|------------------|--------------------
Tea A    | 400       50     | 500        55
Tea B    | 300       80     | 350        90
Tea C    | 200       40     | 280        35

PL = [(500×50) + (350×80) + (280×40)] / [(400×50) + (300×80) + (200×40)] × 100
   = [25000 + 28000 + 11200] / [20000 + 24000 + 8000] × 100
   = 64200 / 52000 × 100 = 123.46

PP = [(500×55) + (350×90) + (280×35)] / [(400×55) + (300×90) + (200×35)] × 100
   = [27500 + 31500 + 9800] / [22000 + 27000 + 7000] × 100
   = 68800 / 56000 × 100 = 122.86

Fisher = √(123.46 × 122.86) = √15173.8 = 123.18

DeFLation (using index numbers):

Real Value = Nominal Value / Price Index × 100

Used to convert nominal GDP to real GDP
Removes inflation effect

Example:

Sri Lanka GDP:
Nominal GDP 2024: Rs. 25 trillion
Price Index 2024 (base 2015): 180

Real GDP (2015 prices) = 25,000 / 180 × 100 = Rs. 13.89 trillion
This allows comparison with GDP in 2015 base year prices

⚡ A/L Key: The difference between Laspeyres and Paasche and knowing when to use each is a classic A/L question. If weights are not changing significantly (or if you’re comparing many periods), Laspeyres is typically used. If consumption patterns change significantly, Paasche may be preferred.

Normal Distribution

Properties:

Bell-shaped, symmetric about the mean
Mean = Median = Mode
Total area under curve = 1 (or 100%)
68% of values within ±1 SD
95% of values within ±2 SD
99.7% of values within ±3 SD

Standard Normal Distribution:

Z = (X - μ) / σ

Where X = value, μ = mean, σ = standard deviation
Z score tells you how many standard deviations from mean

Example:

Monthly revenue of Sri Lankan garment exporters:
Mean = Rs. 50 million, SD = Rs. 8 million

Find probability revenue exceeds Rs. 60 million:
Z = (60 - 50) / 8 = 10/8 = 1.25

From Z table: P(Z > 1.25) = 0.1056
So 10.56% probability revenue exceeds Rs. 60 million

Find probability revenue between Rs. 40M and Rs. 60M:
Z for 40M = (40-50)/8 = -1.25
Z for 60M = 1.25
P(-1.25 < Z < 1.25) = 1 - 2(0.1056) = 0.7888
So 78.88% probability revenue is in this range

⚡ A/L Exam Tip: Z-table values are often provided in the A/L exam paper. Always convert to Z-score first, then use the table. The sign of Z matters — positive Z looks up in positive column, negative Z uses symmetry.

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