Assam Higher Secondary Statistics common mistakes
Assam Higher Secondary Statistics · Understand the traps and how to avoid them.
Plots a 'less than' ogive against class midpoints and reads the median by dropping straight from the curve to the x-axis, and draws a histogram for classes written as 10–19, 20–29 without converting them to exclusive form.
Class marks feel like 'the middle of the class' and therefore look like the natural x-coordinate, while the textbook median is quoted as a curve reading, so the perpendicular step is skipped. Inclusive limits are copied straight from the question without noticing the gap between 19 and 20.
What to do instead
For a 'less than' ogive plot cumulative frequency against upper class boundaries (for 'more than', against lower boundaries), never midpoints. Read the median by going horizontally from N/2 to the curve, then vertically down to the x-axis. Convert inclusive classes to exclusive first (10–19 becomes 9.5–19.5, 20–29 becomes 19.5–29.5) before drawing any histogram or polygon, and note that a histogram cannot be drawn for open-ended classes.
In the grouped-data median formula, substitutes N in place of N/2, picks the class with the highest frequency instead of the class where cumulative frequency first reaches N/2, and uses the stated lower limit l rather than the class boundary.
The ungrouped rule (position of the (n+1)/2 th item) and the grouped rule get mixed together, and because the cumulative frequency column is built in a hurry the 'first cf ≥ N/2' row is misidentified.
What to do instead
Build the cf column first, compute N/2, then circle the first row where cf ≥ N/2 and label it the median class. Apply Median = l + [(N/2 − cf_prev)/f] × h with l = lower boundary of the median class, f = its frequency, cf_prev = cf of the preceding class, h = class width. Final sanity check: the answer must fall inside the median class, otherwise a substitution is wrong.
Computes the combined standard deviation of two groups by averaging σ1 and σ2 (or by averaging the variances), and divides by n instead of n−1 when a sample variance is asked for.
Standard deviation looks like an average, so the arithmetic mean of the two values feels right; the n versus n−1 distinction is memorised as a formula fragment without being tied to population/sample language.
What to do instead
Use the combined mean x̄ = (n1x̄1 + n2x̄2)/(n1+n2), then σ² = [n1(σ1² + d1²) + n2(σ2² + d2²)]/(n1+n2) where d1 = x̄1 − x̄ and d2 = x̄2 − x̄. Never average SDs or variances directly. State the divisor explicitly in your answer and match it to the wording of the question.
Reports the first central moment μ1 as equal to the mean rather than zero, writes Karl Pearson's coefficient of skewness as (Mode − Mean)/SD so the sign reverses, and reads a large β2 as 'positively skewed'.
μ1 and the mean are conflated; the skewness formula is memorised without the direction 'mean minus mode', so the sign carries no meaning in the conclusion; skewness and kurtosis are both filed away as vague 'shape' words.
What to do instead
Anchor three facts: μ1 = 0 always, μ2 = σ², and Sk = (Mean − Mode)/SD, so mean > mode gives a positive value and a tail to the right. Use Sk = 3(Mean − Median)/SD when the mode is ill-defined, and Bowley's Sk = (Q3 + Q1 − 2Q2)/(Q3 − Q1) when quartiles are given. Treat kurtosis only as peakedness: β2 = μ4/μ2², with β2 = 3 mesokurtic, > 3 leptokurtic, < 3 platykurtic (excess γ2 = β2 − 3).
Uses the regression line of y on x to estimate x from a given y, and writes byx = r·σx/σy, i.e. swaps the two coefficients, then checks the work with r = ±√(byx·bxy) without noticing that the product is wrong.
The two formulas look symmetric at a glance and differ only in which σ sits on top, and under exam pressure 'regression of y on x' is silently read as 'regression of x on y'.
What to do instead
Fix the direction first: the line of y on x estimates y, has slope byx = r·σy/σx, and is the one minimising vertical deviations. Use the mnemonic that the subscript order matches the numerator order (byx has σy on top). Both lines pass through (x̄, ȳ), which is a quick check, and r = ±√(byx · bxy) with the sign taken from the coefficients.
Applies P(A∪B) = P(A) + P(B) to events that can occur together, applies P(A∩B) = P(A)P(B) to events that are not independent, and treats 'two cards drawn without replacement' as if the denominator stayed at 52.
The addition and multiplication rules are learnt as two standalone formulas rather than as statements about exclusivity and independence, and the phrase 'without replacement' is never translated into a changing denominator.
What to do instead
Before substituting, write down for each pair whether the events can co-occur (if not, add) and whether P(B|A) = P(B) (if yes, multiply). Use the general rules P(A∪B) = P(A) + P(B) − P(A∩B) and P(A∩B) = P(A)·P(B|A); for without replacement the second denominator drops by one, e.g. two kings = (4/52)(3/51).
Uses the binomial distribution for a rare-event problem such as n = 2000, p = 0.001 because 'there are two outcomes', then quotes the variance as npq; and standardises a normal variable with z = (x − μ)/σ².
The binomial is the first distribution learnt, so it becomes the default; and σ and σ² are used interchangeably while writing quickly under time pressure.
What to do instead
Apply a selection test: fixed n with p not small gives binomial (mean np, variance npq); n large with p very small and np = λ moderate gives Poisson, where mean = variance = λ. For the normal curve always use z = (x − μ)/σ, then table-read; handle 'greater than' as 1 − P(Z ≤ z) and 'between' as a difference of two table values.
Calls any random selection 'stratified sampling', describes systematic sampling as 'picking items at random', and writes that a sample survey is always less accurate than a census.
The definitions overlap in ordinary language, so the distinguishing mechanism is never stored; the accuracy claim comes from equating 'smaller study' with 'worse study' without separating sampling error from non-sampling error.
What to do instead
Define each method by its mechanism: simple random - every unit has an equal chance of selection; stratified - population split into homogeneous strata with a random sample drawn within each; systematic - every kth unit after a random start. On accuracy, state that a well-designed sample usually gives more reliable results than a census because non-sampling errors such as non-response and recording mistakes are smaller, even though sampling error is present.
Ends a test with 'hence H0 is accepted' or 'H0 is proved', uses a one-tailed critical value when the alternative is two-sided, and takes df = n for a single-sample t-test.
Everyday language of proof carries over into hypothesis testing; the tail is decided by the wording of H1 but students look only at the significance level; degrees-of-freedom rules are memorised as bare numbers without the n − 1 logic.
What to do instead
Conclude either 'reject H0' or 'fail to reject H0 at the 5% level'; never 'accept' or 'prove'. Choose the tail from H1 (≠ means two-tailed with α/2 in each tail; > or < means one-tailed). Standard df: one-sample t, n − 1; paired t, number of pairs − 1; two independent samples, n1 + n2 − 2; chi-square goodness of fit, categories − 1 minus estimated parameters; chi-square independence, (r − 1)(c − 1).
Puts current-year quantities into the Laspeyres formula and base-year quantities into Paasche, then lists the index-number tests at random, for example claiming Fisher's ideal index fails the factor-reversal test.
The two formulas differ only in the subscripts attached to price and quantity, and the names give no hint about which weight belongs where; the list of tests is memorised as an unordered jumble.
What to do instead
Write the weights from the name: Laspeyres uses base-year quantities, Σp1q0/Σp0q0 × 100; Paasche uses current-year quantities, Σp1q1/Σp0q1 × 100. Fisher's ideal is the geometric mean √(L × P); it satisfies the time-reversal and factor-reversal tests and the unit test, but not the circular test. Always name the base year and check that each price has the same quantity weight in numerator and denominator.