Exploratory Data Analysis (EDA)

LEARNING SECTION 01

What is exploratory data analysis?

Exploratory Data Analysis (EDA) is the disciplined examination of data before committing to a model. It combines numerical summaries, visual evidence and financial context to identify patterns, unusual observations and assumptions that need investigation.

EDA asks what the observations actually support. It can reveal a heavy-tailed return distribution, a relationship driven by one company, a reporting break or persistent volatility. It generates questions and candidate explanations; a pattern discovered in the same sample is not independent confirmation of a hypothesis.

Learning outcomes: summarise one variable; compare groups and paired variables; inspect several variables together; explore time dependence; read regression diagnostics; and communicate findings with appropriate limitations. This guide follows the univariate, multivariate, visual-diagnostic and time-series themes in the Session 3 teaching material.

LEARNING SECTION 02

How does EDA fit into Financial Analytics?

Begin with the meaning, source and timing of the data. Resolve quality issues and document transformations, then explore the analysis-ready observations. EDA can also send you back to an earlier step: a strange scatter plot may expose a unit mismatch or a duplicated join.

Stage	Main question	Result
Understand	What does each field measure?	Data dictionary and scope
Clean	Are records valid and complete enough for the task?	Validated observations with issue flags
Transform	Which representation fits the question?	Returns, ratios, aligned panel or rescaled features
Explore	What does the dataset reveal?	Charts, summaries, questions and limitations
Model and validate	Does a specified relationship generalise?	Evidence from appropriate estimation and later validation

Do not require every variable to look normal before modelling. Different methods have different assumptions. For example, normal predictors are not a general requirement of linear regression; error properties and functional form are separate issues.

LEARNING SECTION 03

Univariate, bivariate and multivariate exploration

Level	What it examines	Financial application
Univariate	One variable at a time	Return distribution, loan size or sector frequencies
Bivariate	Two variables together	Stock and market returns; default rates by rating band
Multivariate exploration	Several variables jointly	Returns, volatility, turnover and sector together
Time-series exploration	Observations in their time order	Trend, seasonality, dependence and changing variability

The labels overlap in practice. Here “multivariate exploration” means examining several variables jointly. In formal modelling terminology, multiple regression has several predictors and one response, while multivariate regression can have several responses. Use the precise model name when specifying an analysis.

LEARNING SECTION 04

A short glossary before you calculate

Term	Plain meaning
OLS	Ordinary Least Squares: choose coefficients that minimise the sum of squared residuals.
Residual	Actual outcome minus fitted outcome for an observation.
Beta / alpha	Market-return slope / regression intercept. Jensen’s alpha uses excess returns, not unadjusted raw returns.
R²	Share of centred outcome variation explained in the estimation sample for OLS with an intercept. It is not a forecast-accuracy score.
Homoscedasticity	Constant conditional error variance. Changing spread is heteroscedasticity.
ACF / PACF	Lagged linear association / lagged association after controlling for intervening lags.
PCA	Principal Component Analysis: replace correlated variables with orthogonal combinations ordered by captured variance.
Mahalanobis distance	Distance from a mean vector adjusted for variances and correlations.

Keep the sample, units and model next to each statistic. A daily beta, monthly beta and regression on price levels are different calculations.

LEARNING SECTION 05

Profile the dataset before calculating statistics

Check	What to record	Why it matters
Unit of observation	Trade, borrower, security-day or company-year	Determines the meaning of one row
Coverage	Sample dates, population, inclusions and exits	Shows what conclusions the sample can support
Types and units	Numeric, nominal, ordinal, currency and scale	Prevents meaningless arithmetic
Completeness	Missing counts and patterns by group and date	Avoids summaries of a selected surviving sample
Keys and alignment	Unique identifiers, timestamps and paired observations	Prevents duplicate weight and mismatched relationships

A security with no trades can have a stale mark rather than zero risk. A negative earnings-based ratio can be economically meaningful rather than an invalid entry. Carry these domain distinctions into the exploratory report.

LEARNING SECTION 06

Univariate summaries: centre, spread, shape and position

Dimension	Measures	Interpretation
Centre	Mean, median, mode	Typical location under different definitions
Spread	Range, variance, SD and IQR	Overall variability versus central spread
Shape	Skewness and kurtosis	Asymmetry and sensitivity to extreme outcomes
Position	Minimum, quartiles and percentiles	Where an observation lies in the sample
Coverage	Valid count and missing count	How much evidence supports the summary

Mean x̄ = Σxᵢ / n
Sample variance s² = Σ(xᵢ − x̄)² / (n − 1)
IQR = Q3 − Q1

Report units and frequency. A daily-return SD is not an annual SD, and price-level SD is not return volatility. The coefficient of variation, SD/mean, becomes unstable near a zero mean and is usually unsuitable for comparing return series with small or negative means.

LEARNING SECTION 07

Worked case: eight returns, several different messages

Use eight synthetic one-period returns in percentage-point units: −2, −1, 0, 1, 2, 3, 4 and 9. Enter 9 for a 9% return in this example. The sample is intentionally small for hand calculation; it is not sufficient for reliable tail-risk estimation.

Measure	Calculation / result	What it tells us
Mean	16 / 8 = 2.0000 pp	Average one-period simple return
Median	(1 + 2)/2 = 1.5000 pp	Middle of the ordered sample
Sample variance	84 / 7 = 12.0000 pp²	Squared dispersion
Sample SD	√12 = 3.4641 pp	Dispersion in the original return unit
Q1 / Q3	−0.2500 / 3.2500 pp	Inclusive quartile convention
IQR	3.5000 pp	Spread of the central half
Range	9 − (−2) = 11 pp	Distance between extremes

The largest return pulls the mean above the median. That difference suggests asymmetry worth inspecting, but does not prove the shape of a population. If the 9% observation is genuine, retain it; if it is an error, correct it from evidence.

LEARNING SECTION 08

Histogram: look at the distribution, then question the bins

A histogram groups numerical observations into intervals. Its appearance depends on bin width, boundaries and sample size. Compare sensible alternatives before declaring a distribution bimodal, symmetric or heavy-tailed.

Equal-width return bin · pp	Count
−2 ≤ x < 0	2
0 ≤ x < 2	2
2 ≤ x < 4	2
4 ≤ x < 6	1
6 ≤ x < 8	0
8 ≤ x ≤ 10	1

Figure 1 · Equal-width bins make the count heights directly comparable.

With unequal bin widths, use an appropriate density scale if area is intended to represent probability. A density estimate is a smoothed estimate, not extra data. Label whether a histogram shows counts, relative frequency or density.

LEARNING SECTION 09

Box plots: central spread and observations needing review

Inner fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR

For the return sample, the fences are −0.25 − 1.5 × 3.5 = −5.50 and 3.25 + 1.5 × 3.5 = 8.50. The 9% observation lies above the upper fence. Under a common Tukey convention, whiskers end at the most extreme observations within the fences, rather than at the fences themselves.

A box plot is useful for comparing groups, but can hide clusters and sample size. Add individual points or observation counts when groups are small. An outside point is a review flag, not a deletion instruction.

State the quartile convention. Software can use different quantile and whisker definitions. Compare the numerical definitions before interpreting a mismatch between two charts.

LEARNING SECTION 10

Skewness, kurtosis and Q–Q plots

Positive skewness indicates a longer or more influential right tail under the selected measure; negative skewness indicates the opposite direction. Sample skewness can change sharply when the sample is short or an extreme observation is added.

Kurtosis is based on a fourth standardised moment. It is sensitive to extreme values and should not be reduced to a description of peak height. Pearson kurtosis is 3 for a normal distribution; excess kurtosis subtracts 3, giving a normal reference of zero. Excel KURT reports a sample excess-kurtosis estimate.

A Q–Q plot compares sample quantiles with a specified reference distribution. Systematic curvature or tail departures suggest a mismatch, but depend on sample size and the reference. A near-straight normal Q–Q plot is not proof that future tail events follow a normal law.

LEARNING SECTION 11

Categorical univariate analysis and group comparisons

Variable type	Suitable summary	Financial example
Nominal	Frequency and percentage; bar chart	Sector or branch
Ordinal	Ordered frequencies, median category where meaningful	Ordered credit-risk bands
Binary	Counts and proportion with clear coding	Default / no default over a defined horizon

Show the denominator and missing category. A bar chart should not imply equal numerical distances between rating grades. For comparing a numerical outcome across sectors, use grouped summaries and box plots, with group sizes and consistent observation periods.

If 12 of 100 observed borrowers default in a defined one-year sample, the observed default rate is 12%. It is an estimate for that sample, not automatically a forward-looking PD for every borrower. Exposure-weighted and borrower-count-weighted rates answer different questions.

LEARNING SECTION 12

Choose a bivariate display that fits the two variables

Pair	Start with	Investigate
Numeric + numeric	Scatter plot, covariance and correlation	Form, direction, nonlinearity and influential points
Numeric + category	Grouped box plots and summary table	Differences in centre, spread and sample size
Category + category	Cross-tabulation with row/column percentages	Association, base rates and sparse cells

A statistical test, such as a t-test, ANOVA or chi-square test, belongs to a more explicit inferential question with assumptions about sampling, dependence and expected counts. Exploring many groups and then reporting only the strongest p-value can overstate the evidence.

Association is not causation. A scatter plot can reflect confounding, common time trends, reverse direction or sample selection.

LEARNING SECTION 13

Pearson correlation, rank correlation and covariance

Sample covariance sXY = Σ(xᵢ − x̄)(yᵢ − ȳ)/(n − 1)
Pearson r = sXY/(sX × sY)

Pearson correlation measures linear association on a −1 to +1 scale. Covariance retains the product of the variables’ units. Spearman correlation applies correlation to ranks and can capture monotonic relationships; ties need consistent handling.

A zero Pearson correlation does not imply no relationship or independence. For x = −2, −1, 0, 1, 2 and y = x², the Pearson correlation is zero, but y is completely determined by x.

Figure 2 · The five points illustrate a curved relationship; the connecting lines guide the eye.

LEARNING SECTION 14

Beyond Pearson: ranks, categories and group differences

The bivariate workbook extends exploration beyond two continuous variables. Choose a method from the data types and the question; do not assign arbitrary numeric codes to categories and interpret their Pearson correlation.

Variables	Useful exploration	Important qualification
Two numerical variables	Scatter plot; Pearson correlation for linear association; Spearman correlation of ranks for monotonic association	Use average ranks for ties. Spearman is not limited to straight-line patterns.
Category and numerical outcome	Counts, group means/medians and side-by-side box plots	Unequal sample sizes and different dispersions can explain apparent group differences.
Two categories	Contingency table and within-group percentages	Assess association, not a causal effect.

In the workbook’s illustrative 200-customer table, each row contains 100 respondents:

Group	Product A	Product B	Product C	Total
Male	45	25	30	100
Female	30	50	20	100
Total	75	75	50	200

Expected count = row total × column total / grand total
Each expected row = (37.5, 37.5, 25)
χ² = Σ(O − E)²/E = 13.3333; df = (2 − 1)(3 − 1) = 2
p = 0.00127; Cramér’s V = √(13.3333 / 200) = 0.2582

Under the usual independent-observation assumptions, this is evidence of association in the sample. The p-value is not the probability that independence is true. Cramér’s V describes strength without a universal “moderate” cut-off. For salaries across departments, use ANOVA only with its assumptions checked; Welch’s ANOVA can address unequal variances. A significant overall test does not identify every pair that differs.

LEARNING SECTION 15

Worked bivariate case: market sensitivity and R²

Take five synthetic paired excess-return observations, in percentage points. Market X = −2, −1, 0, 1, 2; stock Y = −3, −1, 0, 2, 4. Risk-free returns have already been subtracted for each matching period.

Quantity	Result	Calculation
Mean X / mean Y	0 / 0.4 pp	Five paired observations
Sample covariance	4.2500 pp²	17 / 4
Sample variance of market	2.5000 pp²	10 / 4
OLS slope / beta	1.7000	4.25 / 2.5
Intercept / alpha	0.4000 pp	0.4 − 1.7 × 0
Pearson correlation	0.9948	17 / √292
R² with intercept	0.9897	289 / 292

Fitted stock excess return = 0.4 + 1.7 × market excess return

A market excess return of 1 percentage point gives a fitted stock excess return of 2.1 percentage points. Residuals are actual minus fitted: 0, 0.3, −0.4, −0.1 and 0.2. Their squared sum is 0.30.

This small synthetic case explains the arithmetic, not a usable investment estimate. R² measures in-sample fit; it does not prove causality, pricing-model validity or future performance. R² = r² holds here for a simple OLS regression with an intercept, not for every model.

LEARNING SECTION 16

Workbook case: prices → returns → beta

The bivariate workbook’s Stock Beta sheet supplies 24 illustrative price observations. Read them in the supplied sequence to obtain 23 paired simple returns. These are teaching inputs, not a verified dated market history. The first market move is 4,000 → 4,080 and the stock move is 100 → 103: returns are 2% and 3%.

Step	Excel location / formula	Recomputed result
Market return	E23: =(C23-C22)/C22; fill through E45	23 returns
Stock return	F23: =(D23-D22)/D22; fill through F45	23 returns
Covariance	C53: =COVARIANCE.S(F23:F45,E23:E45)	0.001034876
Market variance	C52: =VAR.S(E23:E45)	0.000385496
Beta, method 1	C57: =C53/C52	2.684529
Beta, method 2	C58: =SLOPE(F23:F45,E23:E45)	2.684529
Beta, method 3	C59: =CORREL(F23:F45,E23:E45)*STDEV.S(F23:F45)/STDEV.S(E23:E45)	2.684529
Intercept	C61: =INTERCEPT(F23:F45,E23:E45)	-0.2382% per period
Correlation / R²	C54 / C62	0.991273 / 0.982623

The three beta methods agree because they use the same aligned observations and consistent sample conventions. Beta measures market sensitivity, not total standalone volatility. A beta below one can coexist with high idiosyncratic risk.

The fitted raw-return line is Rstock = −0.2382% + 2.684529 × Rmarket. A 1 percentage-point change in the market is associated with a fitted 2.6845 percentage-point change in stock return, holding this fitted relationship fixed. This is not a guaranteed move. For a CAPM/Jensen-alpha exercise, first subtract the matched-period risk-free return from both return series.

LEARNING SECTION 17

Multivariate exploration: inspect the joint picture

Use a scatterplot matrix to compare pairs, a correlation heatmap to summarise linear associations and colour or facets to identify groups. A bubble chart can add a third numerical dimension, but specify whether bubble area or radius represents the value.

Tool	What it contributes	Limit
Correlation matrix	Pairwise linear relationships	Can hide nonlinearity and subgroup differences
Scatterplot matrix	Form and unusual pairings	Can become crowded with many variables
Grouped / faceted chart	Relationships within sectors or regimes	Small groups can be unstable
Multiple regression	Conditional association controlling for specified predictors	Specification and omitted variables still matter
PCA / factor / cluster exploration	Lower-dimensional patterns or groups	Scaling and interpretation affect the result

Align every pair by observation identity and time. Pairwise deletion can use different rows for different correlations, producing a matrix that is unsuitable for portfolio optimisation. Inspect coverage and covariance-matrix properties before using it in a risk calculation.

The supplied multivariate workbook uses ten students and five variables: study hours (X1), practice problems (X2), sleep hours (X3), exam score (X4) and stress (X5). This is a deliberately strongly correlated, non-financial teaching dataset. It demonstrates mechanics; it does not estimate relationships in a student population or in financial markets.

Figure 4. Correlations recomputed from 1_Overview!B20:F29. All ten rows are used for every pair.

The matrix is symmetric. Its diagonal entries are one; the diagonal of the covariance matrix instead contains sample variances. Standardising every variable with its sample SD makes the sample covariance matrix of the z-scores equal to this correlation matrix. In finance, the same mechanics support a return covariance matrix, but dates, frequency, missing observations and sampling uncertainty require separate treatment.

LEARNING SECTION 18

PCA and Mahalanobis distance: an intermediate extension

PCA and Mahalanobis distance both use joint variation, but answer different questions. PCA seeks directions that capture variance. Mahalanobis distance asks how unusual an observation is relative to a covariance structure.

Standardise: zᵢⱼ = (xᵢⱼ − x̄ⱼ)/sⱼ
PCA: Rvₖ = λₖvₖ; PC score = zᵀvₖ
Explained-variance share = λₖ / Σλ
Squared Mahalanobis distance: D² = (x − μ)ᵀΣ⁻¹(x − μ)

Component	Eigenvalue, recomputed	Share of total variance
PC1	4.956551	99.1310%
PC2	0.021779	0.4356%
PC3	0.012925	0.2585%
PC4	0.006620	0.1324%
PC5	0.002124	0.0425%
Total	5.000000	100.0000%

These values use the correlation matrix of the workbook’s ten-row input block. The workbook’s PCA tab presents a stored numerical sketch; changing inputs does not automatically recompute that eigenvalue sketch. Compute a fresh eigendecomposition for a new dataset. The large PC1 share reflects the constructed collinearity; it does not establish a causal factor or guarantee predictive usefulness.

For financial yield curves, PCA can explore common level, slope and curvature movements. Fit scaling and component weights on the training period, then apply those weights to later observations. PCA maximises variance, so a low-variance direction can still matter for credit events or another target.

For Mahalanobis distance, an invertible, well-estimated covariance matrix is essential. Nearly redundant variables make inversion unstable. With only ten observations and five variables, treat results as a mechanical demonstration. A χ² reference is not an exact finite-sample calibration for distances computed using the same estimated mean and covariance; robust estimation and appropriate calibration may be needed.

LEARNING SECTION 19

Read multiple regression without overclaiming

Y = α + β1X1 + β2X2 + … + ε
R² = 1 − SSE/SST, for the usual intercept model

A coefficient describes the fitted change associated with one predictor while the specified others are held fixed. Its sign can differ from a simple correlation when other predictors enter the model. Compare units before comparing coefficient magnitudes.

A high R² can reflect overfitting or common trends. Multicollinearity can make individual coefficient estimates unstable even when overall fit looks strong. Adjusted R² accounts for predictor count under its formula, but does not replace out-of-sample validation.

Separate exploratory fitting from confirmation. If EDA suggests a new model, assess it using appropriate later or held-out observations rather than treating its discovery sample as independent evidence.

LEARNING SECTION 20

Visual diagnostics: what should each plot reveal?

Plot	Look for	Possible follow-up
Residuals versus fitted	Curvature or systematic pattern	Reconsider functional form or omitted predictors
Residual spread versus fitted	Fan-shaped or changing spread	Investigate heteroskedasticity and suitable uncertainty estimates
Residuals in time order	Runs, clusters or persistent swings	Investigate serial dependence and changing regimes
Histogram / Q–Q of residuals	Asymmetry and tail mismatch	Review error assumptions relevant to inference
Leverage / influence diagnostics	Observations strongly affecting the fitted model	Verify records; compare estimates with justified diagnostics

An outcome’s histogram is not a residual diagnostic. Residual outliers, high-leverage observations and influential observations are distinct concepts. A visual pattern is a reason to investigate; a flat-looking plot is not proof that every assumption holds.

Normal residuals are not required merely to compute OLS coefficients. Distributional, variance and dependence assumptions affect the validity of particular tests and intervals. Choose diagnostics for the intended use of the model.

LEARNING SECTION 21

Worked visual diagnosis: advertising and sales

The class exercise’s Sheet4 contains 50 paired observations: advertising spend X and sales Y, both in $000. Recomputing from B5:C54 gives:

Fitted sales = 11.397559 + 3.157427 × advertising spend
R² = 0.973184; SSE = 3907.0616 ($000)²
Residual standard error = √(SSE / (50 − 2)) = 9.022035 $000

First observation	Calculation	Result
X / Y	Advertising 4.9; actual sales 32.3	$000
Fitted sales	11.3975586 + 3.1574270 × 4.9	26.868951 $000
Residual	32.3 − 26.868951	+5.431049 $000

Figure 5. Recomputed from the class exercise’s 50 advertising–sales pairs. The Q–Q reference uses residual mean and sample SD.

Read the residual plot for curvature, changing spread and isolated large errors. Read the Q–Q plot for departures from a normal reference, particularly at the ends. Neither chart establishes error independence: inspect residuals in time or collection order separately. Residuals summing to approximately zero follows from OLS with an intercept and is not a model-quality test.

For a candidate advertising spend of 20 ($000), the fitted sales value is 74.5461 ($000). This is a point estimate, not a causal return on advertising or a prediction interval. Observational sales can also reflect prices, seasonality and other omitted drivers.

Excel cell	Enter / fill	Purpose
H21	=INTERCEPT(C5:C54,B5:B54)	Recompute intercept
H22	=SLOPE(C5:C54,B5:B54)	Recompute slope
D5	=$H$21+$H$22*B5; fill to D54	Fitted sales
E5	=C5-D5; fill to E54	Residual
Q5	=NORM.S.INV((ROWS($Q$5:Q5)-0.5)/COUNT($E$5:$E$54)); fill to Q54	Theoretical normal quantile
R5	=SMALL($E$5:$E$54,ROWS($R$5:R5)); fill to R54	Sorted residual for an XY Q–Q plot

LEARNING SECTION 22

Time-series exploration: preserve the order

A time plot reveals information a histogram cannot: when a shock happened, whether volatility clustered, whether values were stale and whether the reporting process changed. Plot price levels and returns separately when both questions matter.

Feature	Meaning	Diagnostic
Trend	Longer-run direction	Time plot and a clearly defined smoother
Seasonality	Repeating pattern at a known period	Seasonal and seasonal-subseries plots
Cycle	Persistent swings without a fixed calendar period	Longer history and economic context
Changing variability	Scale differs across time	Rolling SD or model-based volatility view
Break / anomaly	Unexpected shift or isolated observation	Event and data-quality investigation

An additive representation suits approximately constant-sized seasonal swings. A multiplicative representation can suit proportional swings with a positive series. Neither should be selected from one striking visual segment alone.

LEARNING SECTION 23

Worked time-series case: a moving average is a summary

Consider quarterly revenue, ₹ crore: 100, 120, 110, 140; 110, 130, 120, 150; 120, 140, 130, 160. Each fourth quarter is the local peak, while the overall level rises. Three years illustrate the concept but provide limited evidence for a stable seasonal model.

Figure 3 · A synthetic quarterly pattern, not an empirical company forecast.

Trailing 3-quarter average at t = (Yₜ + Yₜ₋₁ + Yₜ₋₂)/3

The first available three-quarter average is (100 + 120 + 110)/3 = 110.00. The final average is (140 + 130 + 160)/3 = 143.33. A trailing average smooths and lags the series; it is not automatically a forecast.

A centred smoother can use future observations. That may be useful for retrospective description, but it must not enter a real-time forecasting feature as though it were available at the date shown.

LEARNING SECTION 24

ACF, lag plots and stationarity: an exploratory bridge

Conventional sample ACF at lag k:
rₖ = Σₜ₌ₖ₊₁ⁿ (yₜ − ȳ)(yₜ₋ₖ − ȳ) / Σₜ₌₁ⁿ (yₜ − ȳ)²

A lag plot compares observations with their earlier values. The ACF summarises dependence at several lags. For a nonconstant series, the conventional lag-zero ACF is one. ACF estimates on trending levels can show strong persistence even when the period-to-period changes have a different structure.

Weak stationarity concerns a constant mean, finite constant variance and covariance depending on lag rather than calendar time. Plots can suggest violations; unit-root and stationarity tests address specific null hypotheses and have their own assumptions. Returns are not automatically stationary merely because prices were differenced.

The often-used ±1.96/√n bands are approximate pointwise white-noise reference limits. One bar outside is not proof of a real economic relationship; examining many lags creates multiple-testing concerns. Bars inside do not establish independence or absence of nonlinear dependence.

Sample ACF and Pearson correlation between truncated lag pairs are not generally identical: the centring and denominators differ. Document which one you calculate. PACF, ARIMA and formal stationarity methods are developed in the linked time-series guides.

LEARNING SECTION 25

Excel workshop: reproduce the summaries and relationships

On a Returns sheet, enter the eight percentage-point returns in B2:B9. Use ordinary numeric formatting and label the unit. On a Market sheet, enter the five market and stock excess-return pairs in B2:C6. Keep rows paired.

Sheet / cell	Formula	Result / use
Returns!F2	=AVERAGE(B2:B9)	2.0000 pp
Returns!F3	=MEDIAN(B2:B9)	1.5000 pp
Returns!F4	=VAR.S(B2:B9)	12.0000 pp²
Returns!F5	=STDEV.S(B2:B9)	3.4641 pp
Returns!F6	=QUARTILE.INC(B2:B9,1)	−0.2500 pp
Returns!F7	=QUARTILE.INC(B2:B9,3)	3.2500 pp
Returns!F8	=SKEW(B2:B9)	Sample skewness; check sensitivity
Returns!F9	=KURT(B2:B9)	Sample excess kurtosis
Market!F2	=COVARIANCE.S(B2:B6,C2:C6)	4.2500 pp²
Market!F3	=CORREL(B2:B6,C2:C6)	0.9948
Market!F4	=SLOPE(C2:C6,B2:B6)	1.7000
Market!F5	=INTERCEPT(C2:C6,B2:B6)	0.4000 pp
Market!F6	=RSQ(C2:C6,B2:B6)	0.9897
Market!D2; fill down	=$F$5+$F$4*B2	Fitted stock excess return
Market!E2; fill down	=C2-D2	Residual

On a Time sheet, put the 12 quarterly revenues in B2:B13. Enter =AVERAGE(B2:B4) in C4 and fill through C13 for the trailing three-quarter average. =CORREL(B2:B12,B3:B13) is lag-one paired Pearson correlation; label it that way rather than calling it the conventional full-mean sample ACF.

For the conventional lag-one ACF, use =SUMPRODUCT((B3:B13-AVERAGE(B2:B13)),(B2:B12-AVERAGE(B2:B13)))/DEVSQ(B2:B13) on the complete numeric series. The denominator must be non-zero. For categorical summaries, use COUNTIF and a PivotTable with clearly defined denominators.

The formulas above reproduce the article’s synthetic examples. Download the original Session 3 study workbooks in the next section for additional exercises.

LEARNING SECTION 26

Download the Session 3 Excel study notes

Use these original teaching workbooks alongside the article. Begin with the bivariate notes, then move to the intermediate multivariate extension. Their examples are illustrative; they are not a current market-data service.

Bivariate analysis

Covariance, Pearson and Spearman correlation, simple regression, stock beta, contingency tables and group comparisons.

Download bivariate study notes · XLSX

Start here: 2. Covariance → 3. Correlation → 4. Regression → 7. Stock Beta → 8. Categorical Bivariate. Read the stock inputs in observation order.

Multivariate analysis

A ten-row, five-variable teaching dataset with descriptive summaries, matrices, z-scores, Mahalanobis distance, PCA and multiple regression.

Download multivariate study notes · XLSX

Start here: 1_Overview → 3_Covariance → 4_Correlation → 5_Standardization. The 7_PCA eigenvalues are stored approximations, not a live eigenvalue solver; use fresh PCA estimates after changing inputs.

Open in desktop Excel and recalculate before experimenting. Some advanced functions require a compatible Excel version. Keep formulas, data frequency, sample definitions and the order of paired observations consistent. The article’s recalculated examples state their exact input ranges and conventions.

LEARNING SECTION 27

Interactive EDA calculators

Lab 1 calculates descriptive statistics using inclusive quartiles and sample SD. Defaults reproduce the eight-return illustration. Input units remain unchanged; the lab does not annualise returns or automatically remove outliers.

Lab 2 calculates paired Pearson correlation, sample covariance and simple OLS with an intercept. Enter equal-length series in matching observation order. Defaults reproduce the market illustration; the calculator does not establish causality or validate a pricing model.

LEARNING SECTION 28

Turn EDA into a useful analytical conclusion

Report element	Good practice
Question and population	Define the decision, sample, period and units
Coverage and data quality	Quantify missingness, exclusions and unusual records
Main findings	Pair each material statement with a chart or calculation
Alternative explanations	Discuss subgroup composition, trends, events and sensitivity
Modelling implications	Explain the representation and model assumptions to examine next
Limitations and validation	Separate descriptive patterns from causal or predictive claims

Avoid a dashboard of disconnected charts. A concise EDA report explains what was found, why it matters financially and what analysis would distinguish competing explanations. Keep the raw evidence and documented transformations available for reproduction.

LEARNING SECTION 29

Revision questions and answers

What does EDA do before modelling?

It examines distributions, relationships, time structure and data-quality concerns to guide questions and model choices.

What is the mean and sample SD of the eight returns?

Mean 2 percentage points; sample variance 12 pp²; sample SD √12 = 3.4641 pp.

Why is the median lower than the mean?

The large positive return pulls the average upward. Inspect the plot rather than inferring a population shape from eight values.

Is the 9% return an error because it exceeds an IQR fence?

No. It exceeds the 8.5 pp upper fence, which triggers investigation. Its cause must be verified.

Does Pearson r = 0 imply independence?

No. The example y = x² has a deterministic curved relationship with zero Pearson correlation in the symmetric sample.

What is the fitted line in the market case?

Stock excess return = 0.4 + 1.7 × market excess return, with both in percentage points.

Does R² = 98.97% establish forecasting quality?

No. It describes in-sample fit in the five-pair illustration. Validation and model relevance require separate evidence.

What is the final trailing three-quarter revenue average?

(140 + 130 + 160)/3 = ₹143.33 crore.

Can a histogram reveal when volatility increased?

Not by itself. It loses time order; inspect a time plot and a suitable variability diagnostic.

Are residual and outcome histograms interchangeable?

No. Residual diagnostics concern actual minus fitted values and assumptions relevant to the fitted model.

Do all ACF bars inside reference bands prove white noise?

No. The bands are approximate pointwise references and cannot establish independence or rule out nonlinear structure.

What should the EDA report conclude?

State supported patterns, financial implications, alternative explanations, data limitations and the next validation step.

LEARNING SECTION 30

Sources and connected learning

Teaching basis: Prof. V. Ravichandran’s Session 3 Exploratory Data Analysis material. Additional teaching references: Session 3 EDA guide (expanded version), Bivariate Data Analysis, Multivariate Data Analysis, and the advertising–sales exercise in the May 10 class workbook. The article combines original synthetic illustrations with explicitly identified workbook examples; workbook-based figures are recomputed from their input observations.

NIST/SEMATECH · Exploratory Data Analysis — principles, graphical methods and diagnostics.
Hyndman and Athanasopoulos · Time Series Graphics — time plots, seasonal patterns, lag plots and autocorrelation.

Exploratory DataAnalysis (EDA)

See the distribution.Question the relationship.