Q11

1 markMCQSection A

The total area under a standard normal curve is :

Q: The total area under a standard normal curve is :

The correct option is **(A) 1**. **Reason** - Any probability density curve encloses total area equal to 1 (total probability). - The standard normal curve is a probability density, so its total area is 1.

Q: Why is the total area exactly 1 and not $\frac{1}{2}$?

The curve is symmetric about $z = 0$, so each half has area 0.5. The total area (both halves) is therefore $0.5 + 0.5 = 1$, which represents total probability.

Probability Distributions

Standard Normal Distribution

Options

(A)

1

(B)

\sqrt{2}

(C)

2

(D)

\frac{1}{2}

Official Answer

The correct option is (A) 1.

Reason

Any probability density curve encloses total area equal to 1 (total probability).
The standard normal curve is a probability density, so its total area is 1.

standard normal curvetotal areaprobability densitytotal probability onesymmetry about meannormal distributionz-distributionbell curve

Marking Scheme

11 mark: correct option (A) 1.
2No marks for $\sqrt{2}$ , 2 or $\frac{1}{2}$ ; $\frac{1}{2}$ is only half the area.

Hint

Total probability under any density curve is always 1.

Quick Oral Answer

The standard normal curve is a probability density function, so the total area beneath it over the entire real line equals one.

Analysis & Explanation

The area under any probability density function represents total probability, which must equal 1.

Concept

The standard normal curve is the graph of the density $f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$ .
Integrating this density over all z from $-\infty$ to $\infty$ gives exactly 1.

Why the key is right

(A) 1 equals the total probability, which is the total area under the curve.

Why the distractors are wrong

(B) $\sqrt{2}$ appears in the normalising constant $\frac{1}{\sqrt{2\pi}}$ but is not the total area.
(C) 2 would mean total probability exceeds 1, which is impossible.
(D) $\frac{1}{2}$ is only the area on one side of the mean ( $z = 0$ ), not the whole curve; by symmetry each half is 0.5.

Common Mistakes

1Answering $\frac{1}{2}$ , confusing the total area with the area of one half of the symmetric curve.
2Thinking the peak height or normalising constant equals the area.
3Assuming the area depends on the mean or standard deviation; it is always 1.

Interesting Facts

About 68% of the total area under the standard normal curve lies within one standard deviation of the mean, 95% within two, and 99.7% within three - the empirical (68-95-99.7) rule.

Although the normal curve extends infinitely in both directions, the enclosed area converges exactly to 1, a classic example of a finite area under an infinitely long curve.

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Frequently Asked Questions

Why is the total area exactly 1 and not $\frac{1}{2}$ ?

The curve is symmetric about $z = 0$ , so each half has area 0.5. The total area (both halves) is therefore $0.5 + 0.5 = 1$ , which represents total probability.

Does the total area change if the mean or standard deviation changes?

No. For every normal distribution, whatever the mean and standard deviation, the total area under the density curve is always 1 because it represents the certain event.