Descriptive Statistics¶
Q1: When should you use mean, median, or mode?¶
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Use mean when the data is roughly symmetric and free of extreme outliers — it uses every value and is algebraically tractable (you can average averages, for instance).
Use median when the distribution is skewed or has outliers. Income, house prices, and response times are classic examples where the median is far more representative than the mean.
Use mode for categorical data, or when you need the most-common value — e.g., the most-purchased product size, the most-frequent error code.
A practical rule of thumb: if the mean and median diverge significantly, the distribution is skewed and the median is usually the better summary.
Q2: What is variance and standard deviation?¶
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Variance is the average squared deviation from the mean. Squaring serves two purposes: it makes all deviations positive, and it penalises large deviations more heavily.
Standard deviation is the square root of variance, which brings the unit back to the original scale of the data — making it interpretable.
Population variance: σ² = Σ(xᵢ − μ)² / N Sample variance: s² = Σ(xᵢ − x̄)² / (n − 1)
The n − 1 denominator (Bessel's correction) is used for sample variance because using n would systematically underestimate the true population variance. Dividing by n − 1 produces an unbiased estimator.
In practice: standard deviation tells you, on average, how far individual observations fall from the mean. A low SD means values cluster tightly; a high SD means they're spread out.
Q3: What is the difference between population and sample statistics?¶
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A population is the entire group you care about. A sample is a subset drawn from that population.
You almost never have access to the full population, so you compute sample statistics (x̄, s²) and use them to estimate population parameters (μ, σ²).
Key differences in formulas: - Population mean: μ = Σxᵢ / N; Sample mean: x̄ = Σxᵢ / n (same formula, different meaning) - Population variance: divides by N; Sample variance: divides by n − 1
The sample mean is an unbiased estimator of the population mean: E[x̄] = μ. The sample variance with n − 1 is an unbiased estimator of σ².
Interview framing: when a recruiter asks about estimating a parameter, clarify whether you're treating your data as the full population or a sample from a larger one. This affects which formula and which hypothesis test you choose.
Q4: What is a percentile / quantile?¶
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The p-th percentile is the value below which p% of the data falls. The 50th percentile is the median. The 25th and 75th percentiles are the first and third quartiles (Q1 and Q3).
Quantiles are the same concept expressed as fractions (0–1) rather than percentages (0–100). The 0.9 quantile = 90th percentile.
Percentiles are non-parametric — they don't assume a distribution. They're useful for: - Describing spread without being distorted by outliers - Setting SLAs (e.g., "p99 latency < 200 ms") - Reporting rank-based metrics
Q5: What is the interquartile range (IQR)?¶
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The IQR is Q3 − Q1 — the range of the middle 50% of the data. It's a robust measure of spread that is not affected by outliers or the tails of the distribution.
The IQR is used in: - Box plots to define the "box" and whisker lengths - Outlier detection: values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are flagged as outliers (Tukey's fence)
IQR vs standard deviation: SD is sensitive to extreme values; IQR is not. For skewed or contaminated data, IQR is a better description of typical spread.
from scipy.stats import iqr
import numpy as np
data = [1, 2, 3, 4, 5, 6, 7, 100] # outlier at 100
q1, q3 = np.percentile(data, [25, 75])
iqr_val = q3 - q1
lower_fence = q1 - 1.5 * iqr_val
upper_fence = q3 + 1.5 * iqr_val
outliers = [x for x in data if x < lower_fence or x > upper_fence]
print(outliers) # [100]
Q6: What is skewness? What is positive vs negative skew?¶
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Skewness measures the asymmetry of a distribution around its mean.
- Positive skew (right-skewed): the tail extends to the right. Most values are low, but a few are very high. Mean > Median > Mode. Examples: income, house prices, insurance claim sizes.
- Negative skew (left-skewed): the tail extends to the left. Most values are high, but a few are very low. Mean < Median < Mode. Examples: age at retirement in many populations, exam scores with a ceiling effect.
- Zero skew: symmetric, like a normal distribution.
The rule of thumb for skewness values: - |skew| < 0.5: approximately symmetric - 0.5 ≤ |skew| < 1: moderately skewed - |skew| ≥ 1: highly skewed
Why it matters for modelling: many algorithms assume normally distributed residuals. Skewed features may need a log or Box-Cox transform before being fed to a linear model.
Q7: What is kurtosis?¶
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Kurtosis measures the "tailedness" of a distribution — how much probability mass is in the tails compared to a normal distribution.
- Leptokurtic (kurtosis > 3, excess kurtosis > 0): heavy tails, sharp peak. More extreme values than a normal distribution. Financial returns often exhibit this.
- Platykurtic (kurtosis < 3, excess kurtosis < 0): light tails, flat peak. Fewer extremes.
- Mesokurtic (kurtosis = 3, excess = 0): normal distribution.
Most software reports excess kurtosis (Fisher's definition): kurtosis − 3, so a normal distribution has excess kurtosis of 0.
Kurtosis matters when: - Assessing tail risk (finance, insurance) - Checking normality assumptions before applying tests that require it - Understanding model residuals
Q8: What is a box plot and what does it tell you?¶
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A box plot (box-and-whisker plot) summarises the distribution of a dataset with five numbers: minimum (non-outlier), Q1, median, Q3, maximum (non-outlier).
Components: - Box: spans Q1 to Q3 (the IQR) - Line inside box: median - Whiskers: extend to the furthest non-outlier values, typically Q1 − 1.5×IQR and Q3 + 1.5×IQR - Points beyond whiskers: potential outliers, plotted individually
What you can read from a box plot: - Central tendency: where the median sits - Spread: width of the box and whisker length - Skewness: if the median is off-centre within the box, or one whisker is longer - Outliers: dots beyond the whiskers - Comparisons: placing multiple box plots side-by-side compares groups instantly
Box plots are especially useful for comparing distributions across categories (e.g., salary by job title).
Q9: What is the difference between a histogram and a density plot?¶
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A histogram bins continuous data into intervals and counts the number of observations per bin. It is a frequency chart and the shape depends on the chosen bin width.
A density plot (kernel density estimate, KDE) estimates the underlying continuous probability density function. It is smooth, integrates to 1, and is not affected by arbitrary bin-width choices.
Key trade-offs: - Histograms are more interpretable for absolute counts; density plots are better for comparing shapes across groups with different sample sizes. - Histograms can obscure or reveal features depending on bin width; KDE has a bandwidth parameter that similarly needs tuning. - Density plots can extend into impossible regions (e.g., negative values for age) if not bounded properly.
In practice, use a histogram when you want counts or when presenting to a non-technical audience. Use KDE when comparing distributions or overlaying multiple groups.
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import gaussian_kde
data = np.random.normal(size=500)
plt.hist(data, bins=30, density=True, alpha=0.5, label='Histogram')
kde = gaussian_kde(data)
x = np.linspace(data.min(), data.max(), 200)
plt.plot(x, kde(x), label='KDE')
plt.legend()
plt.show()
Q10: What is covariance vs correlation?¶
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Covariance measures the direction of the linear relationship between two variables: positive covariance means they tend to move together; negative means they move in opposite directions. Its magnitude depends on the scale of the variables, so it is not directly comparable across different pairs.
Correlation is standardised covariance — it divides by the product of the standard deviations, yielding a dimensionless value in [−1, 1]. This makes it comparable and interpretable: - +1: perfect positive linear relationship - −1: perfect negative linear relationship - 0: no linear relationship (but there could be a nonlinear one)
Cov(X, Y) = E[(X − μX)(Y − μY)] Corr(X, Y) = Cov(X, Y) / (σX × σY)
When to use each: covariance appears in portfolio theory and PCA; correlation is used for exploratory analysis and feature selection.
Q11: What is the difference between Pearson and Spearman correlation?¶
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Pearson correlation measures the strength of the linear relationship between two continuous variables. It assumes both variables are continuous and roughly normally distributed, and it is sensitive to outliers.
Spearman correlation is a rank-based measure. It converts both variables to ranks and then computes Pearson correlation on those ranks. It captures monotonic relationships (not necessarily linear) and is robust to outliers and non-normal distributions.
When to use which: - Use Pearson when data is continuous, roughly normal, and you expect a linear relationship. - Use Spearman when data is ordinal, heavily skewed, has outliers, or the relationship may be monotonic but nonlinear. - For most EDA pipelines, Spearman is a safer default.
A common interview gotcha: two variables can have Pearson ≈ 0 but Spearman ≠ 0 (nonlinear monotonic relationship), or both ≈ 0 with a clear nonlinear pattern (e.g., a U-shape).
Q12: When is the mean misleading?¶
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The mean is misleading whenever the distribution is skewed, multimodal, or contains outliers — because the mean is pulled toward the tail or toward extreme values and no longer represents a "typical" observation.
Common scenarios: - Income data: a few billionaires inflate the mean; the median income is far lower and more representative of the typical person. - Response times / latency: a small number of very slow requests can inflate the mean significantly. Median or p95/p99 are better. - Multimodal distributions: if you have two clusters (e.g., juniors and seniors in a salary dataset), the mean may fall in a gap between the two groups and represent nobody. - Small samples with one extreme value: one data entry error or genuine outlier can dominate the mean.
The mean is also undefined or infinite for some heavy-tailed distributions (e.g., Cauchy distribution has no finite mean).
Rule of thumb for interviews: always visualise your data before reporting summary statistics. Report median alongside the mean when you suspect skew.
Q13: What is the coefficient of variation?¶
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The coefficient of variation (CV) is the ratio of the standard deviation to the mean, expressed as a percentage:
CV = (σ / μ) × 100%
It measures relative variability — how large the spread is compared to the average. This makes it useful for comparing variability across variables measured on different scales.
Example: if the average height of adults is 170 cm with SD = 10 cm, CV = 5.9%. If the average salary is $70,000 with SD = $20,000, CV = 28.6%. Despite the salary having a larger absolute SD, the CV lets you compare them meaningfully.
Limitations: - Undefined when the mean is zero - Unreliable when the mean is close to zero (CV becomes huge and unstable) - Only meaningful for ratio-scale data with a meaningful zero
CV is often used in quality control, finance (comparing asset volatility), and biological sciences.
Q14: What is the difference between nominal, ordinal, interval, and ratio data?¶
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This is Stevens' typology of measurement scales. Each level adds properties:
Nominal: categories with no intrinsic order. Operations: equality only. Examples: gender, colour, country, product category. You can count; you cannot add or rank.
Ordinal: categories with a meaningful order, but the gaps between ranks are not necessarily equal. Examples: Likert scales (1–5 satisfaction), education level (high school < bachelor < master), star ratings. You can rank; you cannot say "5 stars is 5× better than 1 star."
Interval: ordered with equal, meaningful spacing, but no true zero. Operations: addition, subtraction. Examples: temperature in Celsius/Fahrenheit, calendar year, IQ scores. You can say "30°C is 10° warmer than 20°C," but you cannot say "30°C is twice as hot as 15°C."
Ratio: interval with a true zero that means absence of the quantity. Operations: all arithmetic including multiplication and ratios. Examples: height, weight, time duration, counts, temperature in Kelvin, income. "60 kg is twice 30 kg" is a valid statement.
Why it matters for modelling: - Nominal → one-hot encode or use embeddings - Ordinal → label encode (preserving order) or target encode - Interval/Ratio → use directly; normalise/standardise as needed
Q15: What is the central limit theorem?¶
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The Central Limit Theorem (CLT) states: given a population with any distribution (with finite mean μ and finite variance σ²), the sampling distribution of the sample mean x̄ approaches a normal distribution as sample size n increases — regardless of the shape of the original population distribution.
Specifically: x̄ ~ N(μ, σ²/n) for large enough n.
Key implications: - You can apply z-tests and t-tests to sample means even when the underlying population is not normal, provided n is large enough (often cited as n ≥ 30 as a rough rule). - The standard deviation of the sampling distribution — called the standard error — is σ/√n. This tells you how much the sample mean varies from sample to sample. - As n increases, the standard error decreases, so larger samples give more precise estimates of μ.
This theorem is foundational to nearly all frequentist inference — confidence intervals, hypothesis tests, A/B testing, and regression all rest on it.
import numpy as np
import matplotlib.pyplot as plt
population = np.random.exponential(scale=2, size=100_000) # not normal
sample_means = [np.mean(np.random.choice(population, size=50)) for _ in range(5000)]
plt.hist(sample_means, bins=50, density=True)
plt.title("Sampling Distribution of the Mean (CLT in action)")
plt.show()
# Despite exponential population, sample means look normal