Statistics and Probability Questions for NDA

Measures of Central Tendency

Central tendency tells you where the center of a data set lies. The three main measures are Mean, Median, and Mode. For NDA, arithmetic mean is most frequently tested, especially for grouped data. The relationship between mean, median, and mode (the empirical formula) is a direct one-line question that appears almost every year.

• Arithmetic Mean (AM) for ungrouped data: sum of all values divided by number of values
• For grouped/frequency data: AM = sum of (f times x) divided by sum of f, where f is frequency and x is midpoint
• Median for ungrouped data: middle value when arranged in order; for even n, average of two middle values
• Median for grouped data uses the formula with cumulative frequency and class interval
• Mode is the value with highest frequency; for grouped data use the modal class formula
• Empirical relation: Mode = 3 Median minus 2 Mean (very commonly asked as a fill-in)

Measures of Dispersion

Dispersion tells you how spread out the data is. NDA tests Variance and Standard Deviation most heavily. Remember: Standard Deviation (SD) is the square root of Variance. The Coefficient of Variation (CV) is used to compare two distributions and appears in NDA questions asking which data set is more consistent — lower CV means more consistent.

• Range = Maximum value minus Minimum value; simplest measure of dispersion
• Mean Deviation = Sum of |x - Mean| divided by n
• Variance = Sum of (x - Mean)^2 divided by n for ungrouped data
• For grouped data: Variance = Sum(f * x^2)/Sum(f) - [Sum(f * x)/Sum(f)]^2
• Standard Deviation (SD) = Square root of Variance
• Coefficient of Variation (CV) = (SD / Mean) * 100; lower CV means more consistent data

Basic Probability

Probability measures the likelihood of an event. NDA tests classical probability most often: number of favorable outcomes divided by total outcomes. You must be comfortable with counting techniques (permutations and combinations) to find favorable and total outcomes. Events can be mutually exclusive or independent — knowing which formula to apply is the key skill tested.

• P(A) = Favorable outcomes / Total outcomes; always between 0 and 1
• P(A') = 1 - P(A), where A' is the complement (event not happening)
• Mutually exclusive events: P(A or B) = P(A) + P(B)
• Non-mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B)
• Independent events: P(A and B) = P(A) * P(B)
• Dependent events: P(A and B) = P(A) * P(B given A)

Conditional Probability and Bayes Theorem

Bayes Theorem is the most advanced probability concept tested in NDA. It helps reverse the direction of probability: given that an outcome happened, what was the probable cause? Questions usually involve two or three bags with balls, or two machines producing defective items. The structure is always the same — find the probability of a hypothesis given observed evidence.

• Bayes Theorem is used when you know P(Evidence given Cause) and want P(Cause given Evidence)
• Always identify all possible causes (hypotheses) first
• Compute P(Evidence given each cause) and multiply by prior probability of each cause
• Denominator = total probability = sum of all such products
• These questions appear as 2-3 step calculations — organize in a table to avoid errors
• Total Probability Theorem: P(B) = Sum of P(Ai) * P(B|Ai) for all mutually exclusive Ai

• Using Mode = 3 Median - 2 Mean with wrong signs or wrong order — memorize it exactly as Mode = 3(Median) - 2(Mean), not the reverse
• Confusing Variance with Standard Deviation — NDA questions sometimes ask for one when calculations lead to the other; always check what is asked at the end
• In grouped data, using class boundaries (exact limits) instead of class midpoints as the x value in mean and variance calculations
• Applying the multiplication rule P(A)*P(B) for dependent events — this formula is only for independent events; use conditional probability for dependent ones
• In Bayes theorem questions, forgetting to include the denominator (total probability) and just reporting the numerator as the answer — always divide by the sum of all weighted probabilities

• MeMo aid for empirical relation: Mo = 3Me - 2Ma (Mode = 3 times Median minus 2 times Mean). Say it as Mo-3Me-2Ma like a chant.
• For SD vs Variance: Variance is the Square, SD is the Root. V comes before S in the alphabet, Square root comes after squaring.
• For Bayes: think CAUSE then CLUE — you know the Clue (evidence), find the Cause. Numerator = one cause, Denominator = all causes.
• MICE rule for probability: Mutually exclusive = add, Independent = multiply, Complement = subtract from 1, Exhaustive = total is 1.

• NDA typically asks 2-3 direct formula questions on mean/median/mode from grouped data — these are gift marks if you know the formulas cold. Practice substituting into the median and mode formulas quickly.
• Variance and SD questions often give you raw data of 5-6 numbers. Use the shortcut formula Sum(x^2)/n - (Mean)^2 to save time rather than computing deviations one by one.
• Probability questions on cards, dice, and bags appear almost every paper. Know total outcomes for a standard deck (52), one die (6), two dice (36) by heart — do not waste time recounting.
• Bayes theorem appears as a single high-value conceptual question, usually with exactly 2 bags or 2 machines. If you see the words given that or it turned out to be defective, it is a Bayes problem.
• CV comparison questions (which group is more consistent) are 1-step questions once you have SD and Mean — do not skip them as they look hard but take under 30 seconds.