ISS 2016 Statistics Paper-1 Solution: Question 39 (Kurtosis and Distribution Shape)
Continuing our question-by-question walkthrough of the ISS Statistics Paper-1 archive, today we take up Question 39 from the 2016 paper. This series is meant to be worked through slowly and carefully, so every algebraic and conceptual step is spelled out in full rather than skipped over.
Quick Summary
Topic:Coefficient of Kurtosis (β2) and classification of distribution shape
Question Reference:ISS 2016, Statistics Paper-1, Question 39
Correct Answer:(c) Leptokurtic
The Question, As Asked in the Paper
The burning time of a certain kind of rocket has a probability distribution with mean burning time 5 minutes, variance 0.5 square minutes and coefficient of kurtosis 4. What is the distribution of burning time of these kind of rockets?
(a) Uniform
(b) Mesokurtic
(c) Leptokurtic
(d) Platykurtic
Step 1: Work Out What Is Actually Being Tested
Read the four options carefully before touching any numbers. Option (a), Uniform, is the name of a specific probability distribution (a flat, rectangular one). Options (b), (c) and (d) — Mesokurtic, Leptokurtic, Platykurtic — are not distributions at all. They are shape classifications based on kurtosis, used to describe how peaked or flat a distribution is compared to the normal curve. Since the question gives us a value for the "coefficient of kurtosis," it is asking us to classify the shape, not to identify a named distribution. That immediately tells us the answer must be (b), (c) or (d).
Step 2: Recall the Kurtosis Classification Rule
The coefficient of kurtosis is usually denoted β2. It is defined using the second and fourth central moments of a distribution:
β2 = μ4 / μ22
where μ2 is the variance and μ4 is the fourth central moment. You don't need to compute this from raw data here — the paper has already handed you the value of β2 directly. What you must know by heart is the benchmark value: for the normal distribution, β2 = 3 exactly. Every other distribution is classified by comparing its β2 against this benchmark of 3:
If β2 = 3, the distribution is Mesokurtic (same peakedness as the normal curve).
If β2 > 3, the distribution is Leptokurtic (taller, more peaked in the middle, with heavier/fatter tails than normal).
If β2 < 3, the distribution is Platykurtic (flatter on top, with thinner tails than normal).
An equivalent way to see this is through the "excess kurtosis," γ2 = β2 − 3. When γ2 is positive the curve is leptokurtic; when it is negative, platykurtic; when it is zero, mesokurtic.
Step 3: Apply the Given Value
We are told the coefficient of kurtosis is 4. Comparing this to the benchmark:
β2 = 4 > 3
Equivalently, excess kurtosis γ2 = 4 − 3 = 1, which is positive. Both routes point to the same conclusion: the distribution has a sharper peak and heavier tails than the normal distribution.
Step 4: Eliminate the Distractors
Mean = 5 minutes and variance = 0.5 square minutes were given in the question, but notice that neither value was used anywhere in Steps 2 or 3. That is deliberate. These two numbers only tell you about the location (mean) and spread (variance) of the burning-time distribution — they carry no information about its peakedness. The classification into Mesokurtic, Leptokurtic or Platykurtic depends purely on β2, so the mean and variance here are extra information meant to test whether you can tell which figures are actually relevant to the question being asked.
Pro Tip: The single biggest trap in this question is not the arithmetic — there isn't any — it's the terminology. Students routinely swap "leptokurtic" and "platykurtic" under exam pressure. A reliable way to lock this in for good is a small language anchor: "Lepto" comes from the Greek for thin/slender, which matches a thin, tall, narrow peak; "Platy" means broad/flat, which matches a flat-topped curve. The way this comparison is usually drilled into students is to always convert back to excess kurtosis (β2 − 3) first, so the sign of the number does the remembering for you instead of the name.
Final Answer
Since β2 = 4 > 3, the burning-time distribution is Leptokurtic, which is option (c). This matches the official ISS 2016 Statistics Paper-1 answer key, which also lists (c) as the correct response for Question 39.
Why This Question Matters
This question looks like a numerical problem about rocket burning times, but it is really testing a single piece of bookkeeping: knowing that 3 is the reference value of β2 for the normal distribution, and being able to compare against it without being distracted by unrelated figures. This exact pattern — a wordy setup followed by one number that actually decides the answer — recurs constantly across ISS and other statistics MCQ papers on skewness and kurtosis. Students who drill this kind of "find the one number that matters" reading until it becomes automatic tend to move noticeably faster through the descriptive-statistics section of the paper, which matters a great deal when the exam is timed.
Frequently Asked Questions
What exactly does kurtosis measure?
Kurtosis measures the "tailedness" and peakedness of a probability distribution relative to a normal distribution. A higher kurtosis means more of the variance comes from infrequent extreme deviations (heavier tails and a sharper central peak), while lower kurtosis means the distribution is flatter with thinner tails.
Why is 3 the magic number for kurtosis?
The value 3 is simply the coefficient of kurtosis (β2) of the standard normal distribution, calculated directly from its fourth and second central moments. It became the universal reference point precisely because the normal distribution is the baseline every other distribution's shape is compared against.
Is a leptokurtic distribution always more likely to be "risky" or have outliers?
In practical terms, yes — a leptokurtic distribution produces extreme values more often than a normal distribution with the same variance, which is why heavy-tailed leptokurtic models are commonly used in areas like reliability engineering and financial risk analysis. However, on the exam, you should treat this purely as a mathematical classification based on β2, not infer real-world risk unless the question asks for it.
Could the mean and variance ever be needed in a kurtosis-classification question?
Not for classification itself, since β2 is a standardized ratio that already accounts for spread. Mean and variance would matter only if a follow-up part asked you to actually compute a probability or a confidence interval using the distribution's parameters, which is not what this question asks.
How is coefficient of kurtosis different from coefficient of skewness?
Skewness (often measured via β1 or γ1) describes asymmetry — whether one tail of the distribution is longer than the other. Kurtosis describes peakedness and tail weight along a symmetric axis. A distribution can be symmetric yet still be strongly leptokurtic or platykurtic, so the two measures capture completely different aspects of shape.
What is a real example of a leptokurtic distribution?
The Student's t-distribution with small degrees of freedom, and the Laplace (double exponential) distribution, are classic examples of leptokurtic distributions — both have sharper central peaks and fatter tails than the normal distribution, with β2 greater than 3.
If any step above felt unclear, drop a comment below and we'll walk through it further — and if you know a fellow ISS aspirant working through the same papers, feel free to pass this along to them too.
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