Data Analytics Statistics and Probability for Analytics Syllabus

Statistics and Probability for Analytics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Statistics and Probability for Analytics in Data Analytics, not a summary of it.

1. 1

   Descriptive Statistics

   4 topics

   • Measures of Central Tendency
   • Measures of Dispersion
   • Distribution Shape
   • Percentiles and Quartiles
2. 2

   Probability Fundamentals

   4 topics

   • Probability Rules and Events
   • Conditional Probability and Bayes' Theorem
   • Random Variables
   • Expected Value and Variance
3. 3

   Probability Distributions

   4 topics

   • Normal Distribution
   • Binomial and Poisson Distributions
   • Uniform and Exponential Distributions
   • Central Limit Theorem
4. 4

   Inferential Statistics

   4 topics

   • Sampling Distributions
   • Confidence Intervals
   • Standard Error
   • Estimation Techniques
5. 5

   Hypothesis Testing

   6 topics

   • Null and Alternative Hypotheses
   • p-values and Significance Levels
   • t-tests and z-tests
   • Chi-Square Test
   • ANOVA
   • Type I and Type II Errors
6. 6

   Correlation and Regression Basics

   4 topics

   • Covariance and Correlation
   • Pearson and Spearman Coefficients
   • Simple Linear Regression
   • Interpreting R-squared

Statistics and Probability for Analytics flashcards for Data Analytics

23 of 51 cards from the Statistics and Probability for Analytics deck — real questions with worked answers.

1. What is the arithmetic mean of a dataset, and how is it calculated for a sample of $n$ values?

   The mean is the sum of all values divided by their count: $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$. It is the balance point of the data but is sensitive to outliers.

2. How do you find the median of a dataset, and why is it preferred over the mean for skewed data?

   Sort the values; the median is the middle value (or the average of the two middle values if $n$ is even). It is robust to outliers and skew because it depends only on rank, not magnitude.

3. What is the mode, and can a dataset have more than one?

   The mode is the most frequently occurring value. A dataset can be unimodal, bimodal, or multimodal, and may have no mode if all values occur equally often. It is the only central tendency measure usable for nominal data.

4. For a right-skewed (positively skewed) distribution, what is the typical ordering of mean, median, and mode?

   $\text{mode} < \text{median} < \text{mean}$. The long right tail pulls the mean upward past the median and mode.

5. What is the range of a dataset and its main limitation as a measure of dispersion?

   Range $= x_{\max} - x_{\min}$. It uses only the two extreme values, so it is highly sensitive to outliers and ignores the distribution of all other data points.

6. Write the formulas for population variance and sample variance.

   Population variance: $\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^{2}$. Sample variance: $s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}$, using $n-1$ (Bessel's correction) for an unbiased estimate.

7. Why does sample variance divide by $n-1$ instead of $n$?

   Dividing by $n-1$ (the degrees of freedom) corrects the downward bias that arises because deviations are taken about the sample mean $\bar{x}$ rather than the true mean $\mu$, yielding an unbiased estimator of $\sigma^{2}$.

8. How is the standard deviation related to the variance, and why is it often preferred?

   Standard deviation is the square root of variance: $s = \sqrt{s^{2}}$. It is preferred because it is expressed in the same units as the original data, making it interpretable.

9. What is the coefficient of variation (CV) and when is it useful?

   $CV = \frac{s}{\bar{x}} \times 100\%$. It expresses dispersion relative to the mean, allowing comparison of variability between datasets with different units or very different means.

10. What is the interquartile range (IQR), and how is it used to detect outliers?

    $IQR = Q_3 - Q_1$. Outliers are commonly flagged as values below $Q_1 - 1.5 \times IQR$ or above $Q_3 + 1.5 \times IQR$.

11. Define skewness and interpret positive versus negative values.

    Skewness measures asymmetry of a distribution. Positive skew means a longer right tail (mass concentrated on the left); negative skew means a longer left tail. A symmetric distribution has skewness $\approx 0$.

12. Define kurtosis and distinguish leptokurtic, mesokurtic, and platykurtic distributions.

    Kurtosis measures tail heaviness/peakedness. Mesokurtic has kurtosis like the normal (excess $\approx 0$); leptokurtic has heavy tails and a sharp peak (excess $> 0$); platykurtic has light tails and a flat peak (excess $< 0$).

13. What does a percentile represent, and what does the 90th percentile mean?

    The $p$-th percentile is the value below which $p\%$ of observations fall. The 90th percentile is the value at or below which 90% of the data lie.

14. How do quartiles divide a dataset, and what does each represent?

    Quartiles split ordered data into four equal parts: $Q_1$ (25th percentile), $Q_2$ (50th percentile, the median), and $Q_3$ (75th percentile).

15. How do you convert a raw value to a percentile rank position in an ordered dataset?

    The rank position for the $p$-th percentile can be found as $L = \frac{p}{100}\,(n+1)$, giving the (possibly interpolated) index of the value in the sorted list of $n$ observations.

16. State the three axioms of probability (Kolmogorov's axioms).

    (1) $P(A) \geq 0$ for any event $A$; (2) $P(S) = 1$ for the sample space $S$; (3) for mutually exclusive events, $P(A \cup B) = P(A) + P(B)$.

17. State the general addition rule for the probability of $A$ or $B$.

    $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. The intersection is subtracted to avoid double-counting outcomes in both events.

18. What does it mean for two events to be mutually exclusive, and what is $P(A \cap B)$?

    Mutually exclusive (disjoint) events cannot occur simultaneously, so $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.

19. What is the complement rule in probability?

    $P(A^{c}) = 1 - P(A)$. The probability that an event does not occur equals one minus the probability that it does.

20. Define conditional probability and give its formula.

    The probability of $A$ given $B$ is $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$. It restricts the sample space to outcomes where $B$ occurs.

21. State the multiplication rule for the probability of $A$ and $B$.

    General: $P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$. If $A$ and $B$ are independent, this simplifies to $P(A \cap B) = P(A)\,P(B)$.

22. What condition defines statistical independence of two events?

    Events $A$ and $B$ are independent if $P(A \cap B) = P(A)\,P(B)$, equivalently $P(A \mid B) = P(A)$. Knowing one occurred does not change the probability of the other.

23. State Bayes' theorem.

    $P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}$, where $P(B) = \sum_i P(B \mid A_i)\,P(A_i)$ by the law of total probability. It updates a prior into a posterior using observed evidence.

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