Business Statistics and Data Analysis

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

Descriptive Statistics:

• Mean: Average (sum of values ÷ number of values)
• Median: Middle value when data is arranged in order
• Mode: Most frequently occurring value
• Range: Maximum - Minimum

Measures of Dispersion:

• Variance: Average of squared deviations from mean
• Standard Deviation: √variance (most useful)
• Range: Max - Min

Key Formulas:

Mean (x̄) = Σx / n
Variance (σ²) = Σ(x - x̄)² / n
Standard Deviation (σ) = √[Σ(x - x̄)² / n]

⚡ A/L Exam Tip: Standard deviation is in the same units as data — use this to compare values within the same dataset!

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Business Statistics and Data Analysis — Detailed Study Guide

Data Types and Collection

Types of Data:

Type	Description	Examples	Analysis
Primary	Collected firsthand	Surveys, experiments	More control, specific to needs
Secondary	Already collected by others	Census data, company reports	Faster, cheaper
Quantitative	Numerical	Revenue, number of employees	Statistical analysis
Qualitative	Descriptive, non-numerical	Customer feedback, product types	Coding and categorisation

Data Collection Methods:

Survey Methods:

Method	Description	Pros	Cons
Questionnaire	Written questions	Low cost, large samples	Low response
Interview	Oral questioning	Deep data, high response	Expensive, slow
Telephone survey	Phone-based	Moderate cost	Declining response
Online survey	Digital distribution	Fast, cheap	Sample bias

Sampling Techniques:

Probability Sampling (random selection):

Method	Description	When to Use
Simple random	Every member has equal chance	No pre-existing groups
Systematic	Every kth member	Large populations
Stratified	Random sample from each stratum	Known subgroups
Cluster	Random clusters, all in cluster	Geographic dispersed

Non-Probability Sampling:

Method	Description	Limitation
Convenience	Readily available	Sample bias
Quota	Meet quotas for characteristics	Non-random
Purposive	Selected for specific criteria	Researcher judgment

Sri Lankan Data Sources:

• Department of Census and Statistics
• Central Bank of Sri Lanka
• Sri Lanka Customs
• Line Ministry publications
• World Bank, ADB databases

Frequency Distributions

Constructing Frequency Distributions:

Step 1: Decide on number of classes:

• Generally 5-15 classes
• Too few = lose detail
• Too many = messy

Step 2: Calculate class width:

Class Width = (Max value - Min value) / Number of classes

Step 3: Determine class boundaries:

• Lower limit = minimum value
• Upper limit = lower limit + class width
• Avoid overlapping

Step 4: Tally and count:

• Tally each data point into appropriate class
• Count tallies for frequency

Example - Monthly Sales (in thousands Rs.):

Class	Tally	Frequency
50-60	IIII	4
60-70	IIII IIII	10
70-80	IIII IIII IIII	15
80-90	IIII III	8
90-100	IIII	4
Total		41

Relative Frequency:

Relative Frequency = Class Frequency / Total Frequency

Cumulative Frequency:

Class	Frequency	Cumulative Frequency
50-60	4	4
60-70	10	14
70-80	15	29
80-90	8	37
90-100	4	41

Graphical Presentation:

• Histogram: Bar graph of frequency distribution (no gaps)
• Frequency polygon: Line graph connecting midpoints
• Ogive: Cumulative frequency line graph
• Pie chart: Proportional representation
• Bar chart: Comparing categories

Measures of Central Tendency

The Mean (Average):

Simple Mean:

x̄ = (Σx) / n
where Σx = sum of all values
      n = number of values

Weighted Mean:

x̄w = (Σwx) / (Σw)
where w = weights
      x = values

Example - Weighted Average Cost:

Item	Cost (Rs.)	Quantity	Total
Item A	100	20	2,000
Item B	150	30	4,500
Item C	200	50	10,000
Total		100	16,500

Weighted Mean = 16,500 / 100 = Rs. 165

The Median:

• Middle value when data arranged in order
• For odd n: Middle value
• For even n: Average of two middle values

Example:

• Data: 10, 15, 20, 25, 30
• Median = 20 (middle of 5 values)

Example with even n:

• Data: 10, 15, 20, 25
• Median = (15 + 20) / 2 = 17.5

The Mode:

• Most frequently occurring value
• Can have no mode, one mode, or multiple modes
• Useful for categorical data

Comparing Measures:

Measure	Best For	Limitation
Mean	Interval/ratio data, symmetry	Sensitive to extreme values
Median	Ordinal data, skewed distributions	Ignores magnitude of values
Mode	Modal category, most common item	May not exist or have multiple

Skewness:

• Symmetric: Mean ≈ Median ≈ Mode
• Positively skewed (right): Mean > Median > Mode
• Negatively skewed (left): Mean < Median < Mode

Measures of Dispersion

Why Dispersion Matters:

• Two datasets can have same mean but different spreads
• Mean profit: Rs. 1,00,000
  • Dataset 1: 90,000; 100,000; 110,000 (consistent)
  • Dataset 2: 0; 100,000; 200,000 (risky)

Range:

Range = Maximum value - Minimum value

Variance and Standard Deviation:

Population Variance:

σ² = Σ(x - x̄)² / n

Sample Variance:

s² = Σ(x - x̄)² / (n - 1)

Standard Deviation:

σ = √[Σ(x - x̄)² / n]  (population)
s = √[Σ(x - x̄)² / (n - 1)]  (sample)

Coefficient of Variation:

CV = (Standard Deviation / Mean) × 100%

• Allows comparison of variability between datasets
• Useful for comparing different scales
• Example: CV = 5% vs CV = 15% → first more consistent

Standard Deviation Calculation Step by Step:

Data: 10, 20, 30, 40, 50

1. Calculate mean: (10+20+30+40+50)/5 = 150/5 = 30
2. Calculate deviations: -20, -10, 0, 10, 20
3. Square deviations: 400, 100, 0, 100, 400
4. Sum of squares: 1000
5. Variance: 1000/5 = 200
6. Standard deviation: √200 = 14.14

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🔴 Extended — Deep Study (3mo+)

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Business Statistics and Data Analysis — Complete Notes for A/L Sri Lanka

Correlation and Regression

Correlation Analysis:

• Measures relationship between two variables
• Does NOT imply causation

Scatter Diagram:

• Plot one variable on x-axis, other on y-axis
• Shows direction (positive/negative) and strength of relationship

Correlation Coefficient (r):

r = [nΣxy - (Σx)(Σy)] / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]

Interpreting r:

Value of r	Interpretation
+1.0	Perfect positive correlation
+0.7 to +0.9	Strong positive
+0.4 to +0.6	Moderate positive
+0.1 to +0.3	Weak positive
0.0	No correlation
-0.1 to -0.3	Weak negative
-0.4 to -0.6	Moderate negative
-0.7 to -0.9	Strong negative
-1.0	Perfect negative correlation

Coefficient of Determination (r²):

• Proportion of variation explained by the relationship
• r² = 0.64 → 64% of variation explained
• r² = 0.25 → Only 25% explained

Regression Analysis:

Linear Regression Line:

ŷ = a + bx
where:
b = slope = [nΣxy - (Σx)(Σy)] / [nΣx² - (Σx)²]
a = intercept = ȳ - bx̄

Least Squares Principle:

• Line that minimises sum of squared vertical distances from points to line
• “Best fit” line

Using Regression for Prediction:

1. Plot scatter diagram
2. Verify linear relationship
3. Calculate regression equation
4. Substitute x value to predict y

Example - Sales and Advertising:

Advertising (Rs.‘000)	Sales (Rs.‘000)
10	50
20	70
30	85
40	100
50	115

• Positive correlation expected
• Predict sales at Rs. 35,000 advertising: approximately Rs. 90,000

Assumptions of Regression:

• Linear relationship
• Homoscedasticity (equal variance)
• Normal distribution of errors
• Independence of observations

Index Numbers

What are Index Numbers?:

• Measure of change over time relative to base period
• Base period = 100 (or 1000 for some systems)
• Subsequent periods show percentage change

Simple Price Index:

Price Index = (Price in current year / Price in base year) × 100

Simple Price Index Example:

• Rice price in 2020: Rs. 90/kg
• Rice price in 2023: Rs. 250/kg
• Index = (250 / 90) × 100 = 277.8
• Price increased 177.8% from base year

Laspeyres Index (using base year quantities):

PLI = (ΣP₁Q₀) / (ΣP₀Q₀) × 100
where:
P₁ = current price
P₀ = base price
Q₀ = base quantity

Paasche Index (using current year quantities):

PPI = (ΣP₁Q₁) / (ΣP₀Q₁) × 100
where:
Q₁ = current quantity

Fisher’s Ideal Index (geometric mean):

Fisher = √(Laspeyres × Paasche)

Consumer Price Index (CPI) in Sri Lanka:

• Tracks cost of consumer basket over time
• Released monthly by Department of Census and Statistics
• Base period: 2013 = 100
• Weighted by expenditure shares

Sri Lanka CPI Categories:

• Food and non-alcoholic beverages (~35%)
• Clothing and footwear (~4%)
• Housing, water, electricity, gas (~22%)
• Transport (~12%)
• Communication (~4%)
• Other categories (~23%)

Index Number Applications:

Index	What it Measures	Sri Lankan Source
CPI	Consumer price changes	Census & Statistics Dept
GDP Deflator	All domestic prices	Central Bank
Colombo CPI	Urban consumer prices	Dept of Census
All Item Index	Broad inflation	Central Bank
Producer Price Index	Wholesale prices	Census & Statistics

Real vs. Nominal Values:

Real Value = Nominal Value / Price Index × 100

Example:

• Nominal wage 2020: Rs. 50,000
• CPI 2020: 130 (base 2013=100)
• CPI 2023: 250 (base 2013=100)
• Real wage in 2023 prices: 50,000 × (250/130) = Rs. 96,154

Time Series Analysis

Components of Time Series:

Component	Description	Example
Trend	Long-term movement	GDP growth over decades
Seasonal	Regular pattern within year	Higher rice prices before harvest
Cyclical	Business cycle fluctuations	Expansion and recession
Irregular/Random	Unpredictable	Natural disaster impact

Trend Analysis:

Moving Averages:

• Simple moving average: Average of fixed number of periods
• 3-year moving average: (Year 1 + Year 2 + Year 3) / 3
• Smooths out fluctuations to show trend

Linear Trend (Least squares):

T = a + bt
where:
t = time period
b = slope = [nΣtY - (Σt)(ΣY)] / [nΣt² - (Σt)²]
a = intercept = Ȳ - b*t̄

Forecasting:

• Use trend equation to extrapolate
• Cautions: Past trends may not continue
• External factors can change patterns

Example - Tea Production Trend:

Year	Production (million kg)
2019	300
2020	285
2021	295
2022	310
2023	320

• Calculating trend: Shows slight upward trend
• Projecting 2024: Approximately 330 million kg

Probability and Business Decisions

Basic Probability:

Classical Probability:

P(Event) = Number of favorable outcomes / Total possible outcomes

Example: Rolling a 6 on a die = 1/6

Empirical Probability:

• Based on relative frequency
• P(Event) = Frequency of event / Total frequency

Subjective Probability:

• Personal judgment or expert opinion
• Used when classical/empirical not possible

Probability Rules:

Rule	Formula	Application
Addition	P(A or B) = P(A) + P(B) - P(A and B)	Mutually exclusive vs. non-exclusive
Multiplication	P(A and B) = P(A) × P(B|A)	Dependent vs. independent events
Complement	P(not A) = 1 - P(A)	At least one success

Expected Value:

E(X) = Σ[x × P(x)]

Business Application - Decision Making:

Outcome	Probability	Profit (Rs.)	Expected Profit
High demand	0.3	500,000	150,000
Medium demand	0.5	200,000	100,000
Low demand	0.2	-100,000	-20,000
Expected Value			230,000

Risk Analysis:

• Standard deviation of outcome values
• Coefficient of variation
• Higher CV = more risk

Business Application: Descriptive Statistics for Business

常用 Business Statistics in Sri Lanka:

Sales Analysis:

• Average daily sales
• Sales growth rate
• Market share calculation
• Seasonal variations

Financial Ratios:

• Liquidity ratios (using averages)
• Profitability (using mean profit margins)
• Efficiency (using turnover ratios)

Quality Control:

• Mean, standard deviation of product dimensions
• Control charts using ±3 standard deviations

Sri Lankan Statistical Resources:

• Central Bank Statistical Bulletin
• Department of Census and Statistics publications
• Sri Lanka Customs trade data
• Annual reports of listed companies
• World Bank Development Indicators

⚡ A/L Exam Tip: Statistics questions require practice with numbers. Know your formulas, show your workings, and interpret what your calculations mean in context!

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