The frequency distribution of a research data which is symmetrical in shape…
2014
The frequency distribution of a research data which is symmetrical in shape similar to a normal distribution but center peak is much higher, is
Answer: C. Leptokurtic — Concept: Kurtosis is the fourth standardised moment of a frequency distribution, and in the classical shape classification used in exam statistics it is read…
- A.
Skewed
- B.
Mesokurtic
- C.
Leptokurtic
- D.
Platykurtic
Show answer & explanation
Correct answer: C
Concept: Kurtosis is the fourth standardised moment of a frequency distribution, and in the classical shape classification used in exam statistics it is read as a measure of peakedness — how high and sharp the curve stands at its centre and how heavy its tails are, always relative to the normal curve, which is taken as the reference shape. Peakedness and asymmetry are separate properties: asymmetry is graded by the third moment (skewness), peakedness by the fourth. Under this convention a symmetric curve is named by where its kurtosis sits relative to the normal curve's.
Formally the moment coefficient of kurtosis is β2 = μ4 / μ22, and the excess kurtosis is γ2 = β2 − 3. The normal curve has β2 = 3 and therefore γ2 = 0. In the classical classification the symmetric curve drawn with a higher and sharper central peak than the normal one is the shape labelled β2 > 3, and the flatter and broader one the shape labelled β2 < 3. A caution worth carrying forward: β2 is a property of the whole standardised distribution and is dominated by its tails, so peak height alone does not mathematically determine it — the peakedness picture is the classical shape convention this question is set in, not a theorem.
Application to this question:
The curve is stated to be symmetrical, so its skewness is zero — the third moment carries no information here and an asymmetry label cannot apply.
It is set beside a normal distribution of the same symmetry, so the reference level for its peakedness is β2 = 3, that is γ2 = 0.
Its central peak is described as much higher than that reference. In the classical classification that is precisely the shape filed under β2 > 3, that is positive excess kurtosis γ2 > 0.
A symmetric curve standing above the normal reference on that scale is named leptokurtic, so the distribution described in the stem is leptokurtic.
Contrast with the other values:
Shape name | Kurtosis level | Curve under the classical convention |
|---|---|---|
Skewed | a third-moment label, not a kurtosis level | asymmetric — one tail drawn out further than the other |
Mesokurtic | β2 = 3, γ2 = 0 | the normal reference level itself |
Leptokurtic | β2 > 3, γ2 > 0 | centre higher and narrower than the reference, tails heavier |
Platykurtic | β2 < 3, γ2 < 0 | top flatter and broader than the reference, tails lighter |
Cross-check with the Greek roots the names come from: leptos means slender, so leptokurtic is the slim tall peak; platys means broad and flat, so platykurtic is the low wide curve; mesos means middle, so mesokurtic is the intermediate normal case. Skewness comes from a different family altogether — it grades lopsidedness, not height — which is why a curve stated to be symmetrical is placed on the kurtosis scale rather than the skewness one.