Q.State whether the following statements about the normal distribution are True or False, giving a brief reason for each:

Showing the 12 most recent of 15 on this concept.

1. CBSE 2026Set MARCH1 markMCQ
   Q.In normal distribution, usually which limits include 99% of the observations?

   (a) $\mu \pm 1.96\sigma$

   (b) $\mu \pm 2\sigma$

   (c) $\mu \pm 3\sigma$

   (d) $\mu \pm 2.57\sigma$

   ›Reveal solutionSolution

   About $99\%$ of a normal distribution lies within $\mu\pm2.57\sigma$ (about $2.58\sigma$).

   Standard normal area results: $\mu\pm1.96\sigma$ covers about $95\%$, $\mu\pm3\sigma$ covers about $99.73\%$, and $\mu\pm2.57\sigma$ (more precisely $2.58\sigma$) covers about $99\%$ of the observations. Hence the $99\%$ limits are $\mu\pm2.57\sigma$.

   ✓Final answer

   (d) $\mu\pm2.57\sigma$.

2. CBSE 2026Set MARCH1 markMCQ
   Q.For a normal distribution, approximate value of mean deviation is 20. Which of the following is the value of quartile deviation?

   (a) $\dfrac{25}{3}$

   (b) $\dfrac{32}{3}$

   (c) $24$

   (d) $\dfrac{50}{3}$

   ›Reveal solutionSolution

   Using $QD:MD:SD=10:12:15$, $QD=\tfrac{10}{12}(20)=\tfrac{50}{3}$.

   For a normal distribution the approximate relations are $QD\approx\tfrac23\sigma$ and $MD\approx\tfrac45\sigma$, giving $QD:MD:SD=10:12:15$. Therefore

   $QD=\frac{10}{12}\times MD=\frac{10}{12}\times20=\frac{200}{12}=\frac{50}{3}.$

   (Equivalently $MD=\tfrac45\sigma=20\Rightarrow\sigma=25$, and $QD=\tfrac23\sigma=\tfrac{50}{3}$.)

   ✓Final answer

   (d) $\dfrac{50}{3}$.

3. CBSE 2026Set MARCH1 mark
   Q.What is the shape of normal curve?

   ›Reveal solutionSolution

   The normal curve is a symmetric, single-peaked, bell-shaped curve.

   The graph of the normal distribution is a smooth, bell-shaped curve. It is perfectly symmetric about its mean $\mu$ (where mean $=$ median $=$ mode), rises to a single maximum at $x=\mu$, and falls off equally on both sides. Its two tails extend to $-\infty$ and $+\infty$, getting closer to the horizontal axis without ever meeting it (the axis is an asymptote).

   ✓Final answer

   Bell-shaped and symmetric about the mean.

4. CBSE 2025Set MARCH1 markMCQ
   Q.What is the area under the normal curve to the right hand side of perpendicular line at $X = \mu$ ?

   (a) $0$

   (b) $0.5$

   (c) $1$

   (d) $-0.5$

   ›Reveal solutionSolution

   The normal curve is symmetric about $X = \mu$, so the area on either side of the mean is $0.5$ — option (b).

   GSEB Class-12 Statistics, Normal Distribution:

   A property of the normal distribution is that it is symmetric about its mean $\mu$. The total area under the curve is $1$. The line $X = \mu$ divides this area into two equal parts:

   $\text{Area to the right of } X = \mu = \frac{1}{2} = 0.5$

   ✓Final answer

   (b) $0.5$.

5. CBSE 2025Set MARCH1 markMCQ
   Q.Mean and the first quartile for a normal distribution are 15 and 11 respectively. Which of the following is the value of the third quartile?

   (a) $8$

   (b) $14$

   (c) $19$

   (d) $10$

   ›Reveal solutionSolution

   By the symmetry of the normal curve, $Q_3 = 2\mu - Q_1 = 2(15) - 11 = 19$ — option (c).

   GSEB Class-12 Statistics, Normal Distribution:

   For a normal distribution the quartiles are placed symmetrically about the mean:

   $Q_1 = \mu - 0.6745\,\sigma, \qquad Q_3 = \mu + 0.6745\,\sigma$

   So the mean is exactly midway between $Q_1$ and $Q_3$:

   $\mu - Q_1 = Q_3 - \mu$

   Given $\mu = 15$, $Q_1 = 11$:

   $\mu - Q_1 = 15 - 11 = 4$

   $Q_3 = \mu + 4 = 15 + 4 = 19$

   ✓Final answer

   (c) $19$.

6. CBSE 2025Set MARCH1 mark
   Q.The approximate value of mean deviation for a normal distribution is 12. Find its standard deviation.

   ›Reveal solutionSolution

   With M.D. $\approx \tfrac{4}{5}\sigma = 12$, the standard deviation is $\sigma = \tfrac{5}{4}\times 12 = 15$.

   GSEB Class-12 Statistics, Normal Distribution:

   A property of the normal distribution is that its mean deviation is approximately:

   $\text{M.D.} = \frac{4}{5}\,\sigma$

   Given M.D. $= 12$:

   $12 = \frac{4}{5}\,\sigma$

   $\sigma = \frac{5}{4} \times 12 = 15$

   ✓Final answer

   The standard deviation is $\sigma = 15$.

7. CBSE 2023Set MARCH1 markMCQ
   Q.What is the area under the normal curve to the left hand side of perpendicular line at $X = \mu$?

   (a) (A) $0$

   (b) (B) $-0.5$

   (c) (C) $1$

   (d) (D) $0.5$

   ›Reveal solutionSolution

   By symmetry of the normal curve about the mean, the area to the left of $X=\mu$ is half of the total, i.e. $0.5$. Option (D).

   The total area under a normal curve equals $1$. The curve is perfectly symmetric about the mean $\mu$, so the vertical line at $X=\mu$ divides the total area into two equal halves. Hence the area to the left of $X=\mu$ is

   $\frac{1}{2}=0.5.$

   ✓Final answer

   Correct option: (D) $0.5$.

8. CBSE 2023Set MARCH1 markMCQ
   Q.Which of the following is the approximate value of mean deviation for normal variable?

   (a) (A) $\frac{4}{5}\sigma$

   (b) (B) $\frac{4}{5}\mu$

   (c) (C) $\frac{2}{3}\sigma$

   (d) (D) $\frac{2}{3}\mu$

   ›Reveal solutionSolution

   The mean deviation of a normal variable is approximately $\dfrac{4}{5}\sigma$. Option (A).

   Among the standard measures of dispersion for a normal distribution, the mean deviation (about the mean) is a fixed multiple of the standard deviation. Its approximate value is

   $\text{M.D.}\approx\frac{4}{5}\sigma\;(=0.8\,\sigma),$

   while the quartile deviation is about $\tfrac{2}{3}\sigma$. So the required approximate value is $\tfrac{4}{5}\sigma$.

   ✓Final answer

   Correct option: (A) $\tfrac{4}{5}\sigma$.

9. CBSE 2023Set MARCH1 mark
   Q.What is the shape of normal curve?

   ›Reveal solutionSolution

   The normal curve is a symmetric, bell-shaped, unimodal curve.

   The graph of a normal distribution is a smooth, continuous, bell-shaped curve. It is symmetric about its mean $\mu$, has a single peak (unimodal) at $x=\mu$, and its two tails extend indefinitely on both sides, approaching but never touching the horizontal axis.

   ✓Final answer

   The normal curve is bell-shaped (symmetric and unimodal).

10. CBSE 2022Set MARCH1 markMCQ
    Q.What is the area under the normal curve to the left hand side of perpendicular line $\mu$?

    (a) 0

    (b) 0.5

    (c) 1

    (d) $-0.5$

    ›Reveal solutionSolution

    Because the normal curve is symmetric about $\mu$, the area to the left of the perpendicular at $\mu$ is exactly half the total area, i.e. $\mathbf{0.5}$.

    Reasoning. The total area under a normal curve is $1$, and the curve is perfectly symmetric about its mean $\mu$. The vertical line at $\mu$ therefore splits the area into two equal parts of $0.5$ each.

    ✓Final answer

    Option (b) 0.5.

11. CBSE 2022Set MARCH1 markMCQ
    Q.If the distribution of normal variable is shown as $N(20, 4)$ then which of the following intervals includes 99.73% of observations?

    (a) $(18, 22)$

    (b) $(16, 24)$

    (c) $(14, 26)$

    (d) $(12, 28)$

    ›Reveal solutionSolution

    With $\mu=20$, $\sigma=\sqrt{4}=2$, the interval covering $99.73\%$ is $\mu\pm 3\sigma = 20\pm 6 = \mathbf{(14,26)}$.

    Given: $N(20,4)$ means mean $\mu=20$ and variance $\sigma^2=4$, so $\sigma=2$.

    Step — apply the empirical rule. About $99.73\%$ of observations lie within three standard deviations of the mean:

    $\mu - 3\sigma \ \text{to}\ \mu + 3\sigma = 20 - 6 \ \text{to}\ 20 + 6 = (14,\,26).$

    ✓Final answer

    Option (c) $(14, 26)$.

12. CBSE 2022Set MARCH1 mark
    Q.What percentage of area is covered under the normal curve within the range $\mu - 2\sigma$ to $\mu + 2\sigma$?

    ›Reveal solutionSolution

    The area under the normal curve between $\mu-2\sigma$ and $\mu+2\sigma$ is about $\mathbf{95.45\%}$.

    Explanation (empirical rule). For a normal distribution:

    • $\mu \pm 1\sigma$ covers about $68.27\%$,

    • $\mu \pm 2\sigma$ covers about $95.45\%$,

    • $\mu \pm 3\sigma$ covers about $99.73\%$ of the observations.

    ✓Final answer

    About 95.45% of the area is covered within $\mu - 2\sigma$ to $\mu + 2\sigma$.