# normal distribution

► syllabus content for the Statistics & Probability Topic: SL syllabus (see syllabus section 5.9); HL syllabus (see syllabus section 5.7).

Teaching the normal distribution should include the following components: (1) understand the relationships between the mean, standard deviation and the shape of the normal distribution curve; (2) use appropriate notation and terminology; (3) understand the relationship between the area under the normal curve and probabilities; (4) understand the standardized normal variable z and how it relates to the mean and standard deviation; (5) competent in using a GDC to compute probabilities (area); and values for the mean and/or standard deviation by an inverse process.

Of course, effective use of a GDC is vital in answering exam questions involving a continuous random variable that is distributed normally. But it is certainly possible for a normal distribution question to appear on Paper 1 (no GDC). Consider the following question.

No calculator allowed

Let X be a random variable that is normally distributed with a mean of 40 cm and a standard deviation of 5.4 cm.
(a) On the diagram, shade the region representing $\text{P}\left(X<36\right)$ .

(b) Given that $\text{P}\left(X<36\right)=\text{P}\left(X>a\right)$ , find the value of a.
(c) Given that  $\text{P}\left(X<36\right)=0.23$ (correct to two significant figures), find $\text{P}\left(36 .

click on below to reveal answers for above question

(a)
(b) $a=44\text{\hspace{0.17em}}\text{cm}$ $\text{}$
(c) $\text{P}\left(36

Nevertheless, most exam questions on normal distribution will be on Paper 2 (GDC allowed). The type of normal distribution question that is often challenging for both SL & HL students is the type where the probabilities for two different values of the random variable are given and the student is asked to find the mean and standard deviation. If a student has not practiced this kind of question, then it can be quite tricky for them. Wise use of a GDC can be very helpful on questions like this. See the question below.

GDC allowed

The heights of 10-year old boys at a school follow a normal distribution. It is known that 25% of these 10-year old boys are shorter than 134 cm, and that 5% of them are taller than 152 cm. Find the value of the mean $\mu$ and the value of the standard deviation $\sigma$ .

click on below to reveal answers for above question

$\mu \approx 139\text{\hspace{0.17em}}\text{cm},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \approx 7.76\text{\hspace{0.17em}}\text{cm}$

#### Teaching Materials

EXS_5-9(7)-30v2_normal_dist
Set of 5 exercises covering normal distribution (answers included) appropriate for both SL & HL students

EXS_5-9(7)-40v1_normal_dist
Set of 11 exercises (shown in PDF viewer below) that attempts to cover most of the different types of normal distribution questions that could appear on an exam - both Paper 1 & Paper 2; answers included

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