STATISTICS FOR
BUSINESS (B)
[HSA10a]
COMPLETE EXAM HANDBOOK
Prof. Goedele Dierckx
KU Leuven — Faculty of Economics and Business
Rania El Ghalbzouri
MBA Bridging Programme
Crash Course · Compact Reference · Exam Survival Guide
Rania El Ghalbzouri Page 1
MBA Bridging Programme
,Table of Contents
Chapter 1 Descriptive Statistics
1. Introduction
2. Qualitative Variables
3. Quantitative Variables — Distribution of Outcomes
4. Summary Measures
5. Empirical Distribution Function, Percentiles & Boxplot
6. Transformations
7. Binned Data
8. Relationships Between Variables
9. Relationship Between Two Quantitative Variables
10. Linear Combination of Variables
Chapter 2 Probability Theory and Random Variables
1. Probability — Motivation and Definition
2. Rules for Probabilities
3. Conditional Probability and Independence
4. Discrete Random Variables
5. Mean and Standard Deviation of a Discrete Random Variable
6. The Binomial Distribution
7. Continuous Random Variables
8. The Normal Distribution
9. Normal Approximation for the Binomial Distribution
10. Transformations of Random Variables
11. Sums, Averages and the Central Limit Theorem
Chapter 3 Inferential Statistics
I1. Point Estimators and the Sampling Distribution
I2. Confidence Interval for μ (σ Known)
I3.1 Hypothesis Testing
I3.2 Link Between Confidence Intervals and Hypothesis Tests
I3.3 Critical Values, Critical Region, and Power
Chapter 4 Hypothesis Tests
H1.1 One-Sample t-Test (σ Unknown)
H1.2 Tests for Normality
H2. Comparing Two Means — Paired and Unpaired Data
H3. One-Way ANOVA — Comparing More Than Two Means
H4. Post-Hoc Tests (After ANOVA)
H4.2 t-Test for Pearson Correlation
H5.1 Chi-Square Tests
Chapter 5 Simple Linear Regression
SR.A1 Model Specification of a Simple Linear Regression Model
SR.A2 Assumptions
SR.A3 Non-Linear Models
SR.C1 Decomposition of Variance and R-Square
SR.C2 F-Test
SR.D Checking Assumptions
Rania El Ghalbzouri Page 2
MBA Bridging Programme
,Descriptive Statistics
1 Introduction
Descriptive statistics summarises sample data. The goal is to describe what you observed — not to
generalise to the population (that is inferential statistics).
Two core questions:
• Distribution of outcomes — what values occur and how often?
• Summary measures — what single numbers best characterise the data?
1.1 Qualitative vs. Quantitative Variables
Feature Qualitative (Categorical) Quantitative (Numerical)
Nature of outcomes Categories / labels Numbers with meaningful size
Sub-types Nominal (no order), Ordinal Discrete (counts), Continuous
(ordered) (measurements)
Example Hair colour, political party Income, age, household size
Mean meaningful? No Yes
Graphical tools Bar chart, pie chart Histogram, boxplot, scatterplot
⚠ Common Mistake
Variables coded as numbers are not automatically quantitative. Hair colour coded 1=blond,
2=brown gives a mean of 1.6 — meaningless. Always check whether arithmetic on the values
makes sense.
📘 Exam Recognition
• "Describe the distribution" → provide both a graph and numerical summaries.
• Descriptives describe the sample only — do not generalise to the population.
• A variable that looks numerical but has labelled categories is qualitative.
Rania El Ghalbzouri Page 3
MBA Bridging Programme
,2 Qualitative Variables
2.1 Frequency Tables
A frequency table shows how often each outcome (category) occurs.
Symbol Meaning Formula
k Number of distinct outcomes —
n Total sample size n = n₁ + n₂ + … + nₖ
nᵢ Absolute frequency of outcome i Count of times outcome i occurred
fᵢ Relative frequency of outcome i 𝑛ᵢ
𝑓ᵢ =
𝑛
Key Properties
• Sum of all absolute frequencies: Σ nᵢ = n
• Sum of all relative frequencies: Σ fᵢ = 1 ← useful check
2.2 Mode
The mode is the outcome with the largest frequency. It is the only measure of centre for qualitative
variables — mean and median require numerical ordering.
2.3 Graphical Representations
Graph Description Best used when
Pie / Doughnut chart Sectors proportional to relative Few categories, showing
frequencies proportions
Bar chart (needle graph) x-axis = outcomes; y-axis = absolute Any qualitative variable
or relative frequency
100% stacked bar Bars filled to 100%, showing Comparing across groups
proportional breakdown
💡 Ordering Bars
Nominal variable: order bars from highest to lowest frequency — mode is then visually obvious.
Ordinal variable: keep the natural order of outcomes (do NOT reorder by frequency).
⚠ Misleading Graphs
A y-axis that does not start at 0 exaggerates differences between bars. Always check the y-axis
origin.
Rania El Ghalbzouri Page 4
MBA Bridging Programme
, 📘 Exam Recognition — Qualitative Variables
• Measure of centre for a qualitative variable → the mode.
• Bar chart y-axis not starting at 0 → the graph is misleading.
• Ordinal variable bars reordered by frequency → incorrect; ordinal order must be preserved.
Quiz — Section 2: Qualitative Variables
Q1. Political party preference (Labour, Conservative, Green) is:
a) Quantitative discrete
b) Qualitative nominal
c) Qualitative ordinal
d) Quantitative continuous
Q2. In a sample of 200 people, 80 prefer outcome A. The relative frequency is:
a) 80
b) 0.40
c) 40
d) 1.25
Q3. True or False: You can validly calculate the mean of a nominal variable coded 1, 2, 3.
a) True — numbers are numbers.
b) False — the mean is meaningless.
Q4. For an ordinal variable, bars in a bar chart should be:
a) Ordered from highest to lowest frequency
b) Kept in the natural order of outcomes
Answer Key
Q1: B — Political parties have no natural numerical order → nominal qualitative.
Q2: B — f = = 0.40.
Q3: B — The codes are labels, not quantities. Arithmetic produces meaningless results.
Q4: B — For ordinal variables, natural order must be preserved. Reordering destroys ordinality.
Rania El Ghalbzouri Page 5
MBA Bridging Programme
,3 Quantitative Variables — Distribution of Outcomes
3.1 Needle Graph for Discrete Variables
A discrete quantitative variable takes a countable set of values (e.g., household size: 1, 2, 3, …). A
needle graph (bar chart with bars in numerical order, no gaps) displays the distribution directly.
3.2 Histogram for Continuous Variables
When a variable is continuous (or discrete with many values), individual frequencies are
impractical. Outcomes are grouped into bins and displayed in a histogram.
Steps:
1. Choose bins (contiguous intervals covering all observations).
2. Count observations in each bin (absolute frequency nᵢ); compute relative frequency fᵢ = nᵢ / n.
3. Draw bars — height depends on which y-axis type you choose.
Histogram type y-axis height Area of each bar
Counts nᵢ (absolute frequency) Proportional to count
Proportions fᵢ (relative frequency) Proportional to fᵢ
Density fᵢ ÷ bin width Area = fᵢ (total area = 1)
Density Histogram
𝑓ᵢ 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = =
𝑤ᵢ 𝐵𝑖𝑛 𝑤𝑖𝑑𝑡ℎ
• fᵢ = relative frequency of bin i
• wᵢ = width of bin i
• Area of each bar = fᵢ = proportion of observations in that bin
• Total area under the entire density histogram = 1
⚠ Unequal Bin Widths
With unequal bin widths, a counts histogram is misleading — a wide bin appears tall simply
because it is wide, not because data is more concentrated there. Always use a density
histogram when bin widths differ.
Rania El Ghalbzouri Page 6
MBA Bridging Programme
,3.3 Describing the Shape
Feature Description Visual clue
Symmetric Left and right halves mirror each Bell-shaped or flat
other
Skewed right (+) Long tail to the right; hump on the Mean > Median
left
Skewed left (−) Long tail to the left; hump on the Mean < Median
right
Unimodal One peak One hump
Bimodal Two peaks Two humps
Outliers Observations far from the bulk of Isolated bars far from main group
data
💡 Skewness Memory Aid
The tail points in the direction of skew. Skewed right → tail goes right. Skewed left → tail goes
left.
📘 Exam Recognition — Histograms
• "Describe the distribution" → shape + skewness + unimodal/bimodal + outliers.
• Density histogram: area of bar = proportion of observations in that bin.
• Unequal bin widths + counts histogram → misleading. Correct: density histogram.
• Modal class = bin with the largest density (height), not necessarily the largest count.
Rania El Ghalbzouri Page 7
MBA Bridging Programme
, Quiz — Section 3: Distribution of Outcomes
Q1. In a density histogram, the total area under all bars equals:
a) n (sample size)
b) 1
c) 100
d) Number of bins
Q2. A histogram has a long tail to the right. The distribution is:
a) Symmetric
b) Skewed to the right
c) Skewed to the left
d) Bimodal
Q3. With unequal bin widths, which histogram avoids visual distortion?
a) Counts histogram
b) Proportions histogram
c) Density histogram
Q4. For a right-skewed distribution, which is typically true?
a) Mean < Median
b) Mean = Median
c) Mean > Median
d) Cannot determine without data
Answer Key
Q1: B — Each bar area = fᵢ (relative frequency). All relative frequencies sum to 1.
Q2: B — The tail points right → positive (right) skew.
Q3: C — Density: area = fᵢ, so wider bins are not visually overrepresented.
Q4: C — The long right tail pulls the mean above the median.
Rania El Ghalbzouri Page 8
MBA Bridging Programme
,4 Summary Measures
4.1 Mean
The sample mean x̄ measures the centre of the data.
Sample Mean
1 𝑥̄₁ + 𝑥̄₂ + … + 𝑥̄ₙ
𝑥̄ = 𝛴 𝑥̄ᵢ =
𝑛 𝑛
• xᵢ = the i-th observation
• n = sample size
Weighted Mean (discrete variable with k distinct outcomes)
𝑥̄ = 𝛴 𝑚ⱼ × 𝑓ⱼ
• mⱼ = the j-th distinct outcome value
• fⱼ = nⱼ / n = relative frequency of outcome j
4.2 Variance and Standard Deviation
Variance and SD measure how dispersed observations are around the mean.
Sample Variance
1
𝑠² = 𝛴 (𝑥̄ᵢ − 𝑥̄)²
𝑛 − 1
• (xᵢ − x̄)² = squared deviation of observation i from the mean
• We divide by n−1 (not n) to make s² an unbiased estimator of the population variance σ²
• Unit: square of the original variable's unit
Sample Standard Deviation
1
𝑠 = √ 𝛴 (𝑥̄ᵢ − 𝑥̄)²
𝑛 − 1
• s ≥ 0 always; s = 0 only when all observations are identical
• Unit: same as the original variable (unlike variance)
• Larger s → more dispersed data
Variance — Discrete Variable with k Distinct Outcomes
𝑠² = 𝛴 (𝑚ⱼ − 𝑥̄)² × 𝑓ⱼ
Rania El Ghalbzouri Page 9
MBA Bridging Programme
, ⚠ Why n − 1?
Dividing by n would systematically underestimate the true population variance σ². Dividing by
n−1 corrects this bias and makes s² unbiased.
4.3 Median
The median is the middle value when observations are sorted from smallest to largest.
Median
n odd: median = observation at position (n+1)/2
n even: median = average of observations at positions n/2 and (n/2)+1
4.4 Mode
• Discrete data: the value with the highest frequency.
• Continuous data (binned): modal class = bin with the highest density. Mode = midpoint of
that bin.
4.5 Range and IQR
Range and Interquartile Range
𝑅𝑎𝑛𝑔𝑒 = 𝑥̄𝑚𝑎𝑥 − 𝑥̄𝑚𝑖𝑛
𝐼𝑄𝑅 = 𝑄₃ − 𝑄₁
• Range = total spread of all data (sensitive to outliers)
• IQR = spread of the middle 50% of data (after ordering)
• Q₁ = 25th percentile; Q₃ = 75th percentile
4.6 Skewness and Kurtosis
Measure What it measures Interpretation
Skewness Asymmetry of distribution = 0 (symmetric), > 0 (right-
skewed), < 0 (left-skewed)
Kurtosis Fatness of tails vs. normal = 0 (normal), > 0 (fatter tails)
distribution
4.7 When to Use Which Measure
Situation Preferred Centre Preferred Spread Why
Symmetric, no outliers Mean x̄ Standard Efficient for symmetric data
deviation s
Skewed distribution Median IQR Mean pulled toward tail;
median is not
Outliers present Median IQR Mean and SD are heavily
influenced; median/IQR are
robust
Rania El Ghalbzouri Page 10
MBA Bridging Programme
BUSINESS (B)
[HSA10a]
COMPLETE EXAM HANDBOOK
Prof. Goedele Dierckx
KU Leuven — Faculty of Economics and Business
Rania El Ghalbzouri
MBA Bridging Programme
Crash Course · Compact Reference · Exam Survival Guide
Rania El Ghalbzouri Page 1
MBA Bridging Programme
,Table of Contents
Chapter 1 Descriptive Statistics
1. Introduction
2. Qualitative Variables
3. Quantitative Variables — Distribution of Outcomes
4. Summary Measures
5. Empirical Distribution Function, Percentiles & Boxplot
6. Transformations
7. Binned Data
8. Relationships Between Variables
9. Relationship Between Two Quantitative Variables
10. Linear Combination of Variables
Chapter 2 Probability Theory and Random Variables
1. Probability — Motivation and Definition
2. Rules for Probabilities
3. Conditional Probability and Independence
4. Discrete Random Variables
5. Mean and Standard Deviation of a Discrete Random Variable
6. The Binomial Distribution
7. Continuous Random Variables
8. The Normal Distribution
9. Normal Approximation for the Binomial Distribution
10. Transformations of Random Variables
11. Sums, Averages and the Central Limit Theorem
Chapter 3 Inferential Statistics
I1. Point Estimators and the Sampling Distribution
I2. Confidence Interval for μ (σ Known)
I3.1 Hypothesis Testing
I3.2 Link Between Confidence Intervals and Hypothesis Tests
I3.3 Critical Values, Critical Region, and Power
Chapter 4 Hypothesis Tests
H1.1 One-Sample t-Test (σ Unknown)
H1.2 Tests for Normality
H2. Comparing Two Means — Paired and Unpaired Data
H3. One-Way ANOVA — Comparing More Than Two Means
H4. Post-Hoc Tests (After ANOVA)
H4.2 t-Test for Pearson Correlation
H5.1 Chi-Square Tests
Chapter 5 Simple Linear Regression
SR.A1 Model Specification of a Simple Linear Regression Model
SR.A2 Assumptions
SR.A3 Non-Linear Models
SR.C1 Decomposition of Variance and R-Square
SR.C2 F-Test
SR.D Checking Assumptions
Rania El Ghalbzouri Page 2
MBA Bridging Programme
,Descriptive Statistics
1 Introduction
Descriptive statistics summarises sample data. The goal is to describe what you observed — not to
generalise to the population (that is inferential statistics).
Two core questions:
• Distribution of outcomes — what values occur and how often?
• Summary measures — what single numbers best characterise the data?
1.1 Qualitative vs. Quantitative Variables
Feature Qualitative (Categorical) Quantitative (Numerical)
Nature of outcomes Categories / labels Numbers with meaningful size
Sub-types Nominal (no order), Ordinal Discrete (counts), Continuous
(ordered) (measurements)
Example Hair colour, political party Income, age, household size
Mean meaningful? No Yes
Graphical tools Bar chart, pie chart Histogram, boxplot, scatterplot
⚠ Common Mistake
Variables coded as numbers are not automatically quantitative. Hair colour coded 1=blond,
2=brown gives a mean of 1.6 — meaningless. Always check whether arithmetic on the values
makes sense.
📘 Exam Recognition
• "Describe the distribution" → provide both a graph and numerical summaries.
• Descriptives describe the sample only — do not generalise to the population.
• A variable that looks numerical but has labelled categories is qualitative.
Rania El Ghalbzouri Page 3
MBA Bridging Programme
,2 Qualitative Variables
2.1 Frequency Tables
A frequency table shows how often each outcome (category) occurs.
Symbol Meaning Formula
k Number of distinct outcomes —
n Total sample size n = n₁ + n₂ + … + nₖ
nᵢ Absolute frequency of outcome i Count of times outcome i occurred
fᵢ Relative frequency of outcome i 𝑛ᵢ
𝑓ᵢ =
𝑛
Key Properties
• Sum of all absolute frequencies: Σ nᵢ = n
• Sum of all relative frequencies: Σ fᵢ = 1 ← useful check
2.2 Mode
The mode is the outcome with the largest frequency. It is the only measure of centre for qualitative
variables — mean and median require numerical ordering.
2.3 Graphical Representations
Graph Description Best used when
Pie / Doughnut chart Sectors proportional to relative Few categories, showing
frequencies proportions
Bar chart (needle graph) x-axis = outcomes; y-axis = absolute Any qualitative variable
or relative frequency
100% stacked bar Bars filled to 100%, showing Comparing across groups
proportional breakdown
💡 Ordering Bars
Nominal variable: order bars from highest to lowest frequency — mode is then visually obvious.
Ordinal variable: keep the natural order of outcomes (do NOT reorder by frequency).
⚠ Misleading Graphs
A y-axis that does not start at 0 exaggerates differences between bars. Always check the y-axis
origin.
Rania El Ghalbzouri Page 4
MBA Bridging Programme
, 📘 Exam Recognition — Qualitative Variables
• Measure of centre for a qualitative variable → the mode.
• Bar chart y-axis not starting at 0 → the graph is misleading.
• Ordinal variable bars reordered by frequency → incorrect; ordinal order must be preserved.
Quiz — Section 2: Qualitative Variables
Q1. Political party preference (Labour, Conservative, Green) is:
a) Quantitative discrete
b) Qualitative nominal
c) Qualitative ordinal
d) Quantitative continuous
Q2. In a sample of 200 people, 80 prefer outcome A. The relative frequency is:
a) 80
b) 0.40
c) 40
d) 1.25
Q3. True or False: You can validly calculate the mean of a nominal variable coded 1, 2, 3.
a) True — numbers are numbers.
b) False — the mean is meaningless.
Q4. For an ordinal variable, bars in a bar chart should be:
a) Ordered from highest to lowest frequency
b) Kept in the natural order of outcomes
Answer Key
Q1: B — Political parties have no natural numerical order → nominal qualitative.
Q2: B — f = = 0.40.
Q3: B — The codes are labels, not quantities. Arithmetic produces meaningless results.
Q4: B — For ordinal variables, natural order must be preserved. Reordering destroys ordinality.
Rania El Ghalbzouri Page 5
MBA Bridging Programme
,3 Quantitative Variables — Distribution of Outcomes
3.1 Needle Graph for Discrete Variables
A discrete quantitative variable takes a countable set of values (e.g., household size: 1, 2, 3, …). A
needle graph (bar chart with bars in numerical order, no gaps) displays the distribution directly.
3.2 Histogram for Continuous Variables
When a variable is continuous (or discrete with many values), individual frequencies are
impractical. Outcomes are grouped into bins and displayed in a histogram.
Steps:
1. Choose bins (contiguous intervals covering all observations).
2. Count observations in each bin (absolute frequency nᵢ); compute relative frequency fᵢ = nᵢ / n.
3. Draw bars — height depends on which y-axis type you choose.
Histogram type y-axis height Area of each bar
Counts nᵢ (absolute frequency) Proportional to count
Proportions fᵢ (relative frequency) Proportional to fᵢ
Density fᵢ ÷ bin width Area = fᵢ (total area = 1)
Density Histogram
𝑓ᵢ 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = =
𝑤ᵢ 𝐵𝑖𝑛 𝑤𝑖𝑑𝑡ℎ
• fᵢ = relative frequency of bin i
• wᵢ = width of bin i
• Area of each bar = fᵢ = proportion of observations in that bin
• Total area under the entire density histogram = 1
⚠ Unequal Bin Widths
With unequal bin widths, a counts histogram is misleading — a wide bin appears tall simply
because it is wide, not because data is more concentrated there. Always use a density
histogram when bin widths differ.
Rania El Ghalbzouri Page 6
MBA Bridging Programme
,3.3 Describing the Shape
Feature Description Visual clue
Symmetric Left and right halves mirror each Bell-shaped or flat
other
Skewed right (+) Long tail to the right; hump on the Mean > Median
left
Skewed left (−) Long tail to the left; hump on the Mean < Median
right
Unimodal One peak One hump
Bimodal Two peaks Two humps
Outliers Observations far from the bulk of Isolated bars far from main group
data
💡 Skewness Memory Aid
The tail points in the direction of skew. Skewed right → tail goes right. Skewed left → tail goes
left.
📘 Exam Recognition — Histograms
• "Describe the distribution" → shape + skewness + unimodal/bimodal + outliers.
• Density histogram: area of bar = proportion of observations in that bin.
• Unequal bin widths + counts histogram → misleading. Correct: density histogram.
• Modal class = bin with the largest density (height), not necessarily the largest count.
Rania El Ghalbzouri Page 7
MBA Bridging Programme
, Quiz — Section 3: Distribution of Outcomes
Q1. In a density histogram, the total area under all bars equals:
a) n (sample size)
b) 1
c) 100
d) Number of bins
Q2. A histogram has a long tail to the right. The distribution is:
a) Symmetric
b) Skewed to the right
c) Skewed to the left
d) Bimodal
Q3. With unequal bin widths, which histogram avoids visual distortion?
a) Counts histogram
b) Proportions histogram
c) Density histogram
Q4. For a right-skewed distribution, which is typically true?
a) Mean < Median
b) Mean = Median
c) Mean > Median
d) Cannot determine without data
Answer Key
Q1: B — Each bar area = fᵢ (relative frequency). All relative frequencies sum to 1.
Q2: B — The tail points right → positive (right) skew.
Q3: C — Density: area = fᵢ, so wider bins are not visually overrepresented.
Q4: C — The long right tail pulls the mean above the median.
Rania El Ghalbzouri Page 8
MBA Bridging Programme
,4 Summary Measures
4.1 Mean
The sample mean x̄ measures the centre of the data.
Sample Mean
1 𝑥̄₁ + 𝑥̄₂ + … + 𝑥̄ₙ
𝑥̄ = 𝛴 𝑥̄ᵢ =
𝑛 𝑛
• xᵢ = the i-th observation
• n = sample size
Weighted Mean (discrete variable with k distinct outcomes)
𝑥̄ = 𝛴 𝑚ⱼ × 𝑓ⱼ
• mⱼ = the j-th distinct outcome value
• fⱼ = nⱼ / n = relative frequency of outcome j
4.2 Variance and Standard Deviation
Variance and SD measure how dispersed observations are around the mean.
Sample Variance
1
𝑠² = 𝛴 (𝑥̄ᵢ − 𝑥̄)²
𝑛 − 1
• (xᵢ − x̄)² = squared deviation of observation i from the mean
• We divide by n−1 (not n) to make s² an unbiased estimator of the population variance σ²
• Unit: square of the original variable's unit
Sample Standard Deviation
1
𝑠 = √ 𝛴 (𝑥̄ᵢ − 𝑥̄)²
𝑛 − 1
• s ≥ 0 always; s = 0 only when all observations are identical
• Unit: same as the original variable (unlike variance)
• Larger s → more dispersed data
Variance — Discrete Variable with k Distinct Outcomes
𝑠² = 𝛴 (𝑚ⱼ − 𝑥̄)² × 𝑓ⱼ
Rania El Ghalbzouri Page 9
MBA Bridging Programme
, ⚠ Why n − 1?
Dividing by n would systematically underestimate the true population variance σ². Dividing by
n−1 corrects this bias and makes s² unbiased.
4.3 Median
The median is the middle value when observations are sorted from smallest to largest.
Median
n odd: median = observation at position (n+1)/2
n even: median = average of observations at positions n/2 and (n/2)+1
4.4 Mode
• Discrete data: the value with the highest frequency.
• Continuous data (binned): modal class = bin with the highest density. Mode = midpoint of
that bin.
4.5 Range and IQR
Range and Interquartile Range
𝑅𝑎𝑛𝑔𝑒 = 𝑥̄𝑚𝑎𝑥 − 𝑥̄𝑚𝑖𝑛
𝐼𝑄𝑅 = 𝑄₃ − 𝑄₁
• Range = total spread of all data (sensitive to outliers)
• IQR = spread of the middle 50% of data (after ordering)
• Q₁ = 25th percentile; Q₃ = 75th percentile
4.6 Skewness and Kurtosis
Measure What it measures Interpretation
Skewness Asymmetry of distribution = 0 (symmetric), > 0 (right-
skewed), < 0 (left-skewed)
Kurtosis Fatness of tails vs. normal = 0 (normal), > 0 (fatter tails)
distribution
4.7 When to Use Which Measure
Situation Preferred Centre Preferred Spread Why
Symmetric, no outliers Mean x̄ Standard Efficient for symmetric data
deviation s
Skewed distribution Median IQR Mean pulled toward tail;
median is not
Outliers present Median IQR Mean and SD are heavily
influenced; median/IQR are
robust
Rania El Ghalbzouri Page 10
MBA Bridging Programme
