IB Mathematics Analysis and Approaches Topic SL…

The Normal Distribution: Why So Many Things Cluster Around An Average

Measure the heights of a thousand adults, the repeated readings of one careful experiment, or the exam scores of a whole cohort.
Plot how often each value shows up and the same picture keeps appearing: a single hump near the middle, thinning out symmetrically on both sides.
That recurring shape is the normal distribution, and it is the most important distribution in statistics.
It keeps appearing because whenever a quantity is the sum of many small independent effects, the pile-up of those effects is approximately normal.
That is the central limit theorem, and it is why the normal curve turns up far beyond the cases anyone designed it for.

The Shape And Its Two Controls, $\mu$ And $\sigma$

A normal curve is a smooth, symmetric bell whose single highest point sits directly above the mean.
Because it is perfectly symmetric, the mean, median and mode all land at that same central value.
The mean $\mu$ sets the centre: Change $\mu$ and the whole bell slides left or right without changing shape.
The standard deviation $\sigma$ sets the width: A small $\sigma$ gives a tall narrow bell; a large $\sigma$ gives a short wide one, because the same total area is spread over more values.
We write $X \sim N(\mu,\, \sigma^{2})$, where the second entry is the variance $\sigma^{2}$, not the standard deviation.

Note

The total area under any normal curve is exactly 1, because the variable must take some value.
A probability is therefore an area under the curve, and the tails approach the axis without ever touching it.

The 68 95 99.7 Rule: A Quick Feel For The Spread

Because the shape is always the same once distance is measured in standard deviations, fixed percentages of the data fall within fixed numbers of $\sigma$ from the mean.
This lets you sanity-check any normal answer without a calculator.
1. About 68% of the data lies within $\mu \pm \sigma$.
2. About 95% lies within $\mu \pm 2\sigma$.
3. About 99.7% lies within $\mu \pm 3\sigma$.

Example

Adult male heights are roughly normal with mean 175 cm and standard deviation 7 cm.
So about 68% of men are between 168 cm and 182 cm, one standard deviation either side.
And about 95% are between 161 cm and 189 cm, two standard deviations either side.

Reading Probabilities As Areas

Every normal probability question is really an area question.
$P(X < x)$ is the area to the left of $x$, $P(X > x)$ is the area to the right, and $P(a < X < b)$ is the strip in between.
Because the curve is continuous, $P(X < x)$ and $P(X \le x)$ are equal, so the single point makes no difference.

Exam technique

The normal distribution has no by-hand formula in the formula booklet for these areas, so every probability comes from your GDC.
The booklet does list the standardized variable $z = \dfrac{x - \mu}{\sigma}$ under Topic 4, but you only need that in subtopic 4.12.
At this stage you always work directly from $\mu$ and $\sigma$, so keep the calculator in the driving seat.

Normal Calculations On The GDC: Two Commands

One command turns values into probabilities, and the other turns probabilities back into values.
Knowing which one a question wants is half the battle.

You are given	You want	GDC command
mean, SD and boundary values	a probability (area)	normalcdf
mean, SD and a probability	a boundary value	invNorm

For normalcdf you enter the lower bound, upper bound, mean and standard deviation, using a huge number for a missing one-sided bound:

Event	GDC entry
P(X < a)	normalcdf(-1E99, a, mean, SD)
P(X > a)	normalcdf(a, 1E99, mean, SD)
P(a < X < b)	normalcdf(a, b, mean, SD)

Common Mistake

The GDC wants the standard deviation $\sigma$, but the notation $N(\mu, \sigma^{2})$ gives the variance, so square-root it first if needed.
For "greater than" use the right tail, or take the complement $1 - P(X < a)$.

Example question

The mass $X$ of an apple is normally distributed with mean $\mu = 50$ g and standard deviation $\sigma = 10$ g. Find (a) $P(X < 60)$ and (b) $P(45 < X < 65)$.

Solution

a): area to the left of 60, so $P(X < 60) = \texttt{normalcdf}(-1\text{E}99, 60, 50, 10)$.
$= 0.841$ (3 s.f.).
(b): area of the strip, so $P(45 < X < 65) = \texttt{normalcdf}(45, 65, 50, 10)$.
$= 0.625$ (3 s.f.).

Inverse Normal: From An Area Back To A Value

Sometimes you know the probability and want the value that produces it, such as the 90th percentile or the mark that only the top 5% beat.
This is the inverse normal, and the key rule never changes: invNorm works from the area to the LEFT of the value.

Common Mistake

invNorm always expects the left-hand area, the probability of being below the value.
For "the top 10%", the value has 90% below it, so enter 0.90, not 0.10.
Sketch the curve and shade the area first, which stops almost every inverse-normal error.

Example question

Test scores are normally distributed with mean 100 and standard deviation 15. (a) Find the score below which 90% of candidates lie. (b) Find the score that only the top 5% exceed.

Solution

(a): 90% lies below, so the value is $\texttt{invNorm}(0.90, 100, 15)$.
$= 119$ (3 s.f.).
b): the top 5% exceed it, so 95% lie below it: $\texttt{invNorm}(0.95, 100, 15)$.
$= 125$ (3 s.f.).

See The Area Move As The Boundary Slides

The link between a boundary value and a probability is easiest to feel by watching the shaded area grow and shrink as the boundary moves across the curve.
The whole subtopic reduces to one idea: a normal probability is an area under the bell, and your GDC measures that area from $\mu$ and $\sigma$.

Active recall

In the notation $X \sim N(\mu, \sigma^{2})$, is the second number the standard deviation or the variance, and why does that matter on the GDC?
State the approximate percentage of data within one, two and three standard deviations of the mean.
Which GDC command turns boundary values into a probability, and which turns a probability into a boundary value?
To find the value that the top 20% of a normal distribution exceeds, what left-hand area do you enter into invNorm?
Explain why $P(X < x)$ and $P(X \le x)$ are equal for a normal distribution.

The Normal Distribution: Why So Many Things Cluster Around An Average

Measure the heights of a thousand adults, the repeated readings of one careful experiment, or the exam scores of a whole cohort.
Plot how often each value shows up and the same picture keeps appearing: a single hump near the middle, thinning out symmetrically on both sides.
That recurring shape is the normal distribution, and it is the most important distribution in statistics.
It keeps appearing because whenever a quantity is the sum of many small independent effects, the pile-up of those effects is approximately normal.
That is the central limit theorem, and it is why the normal curve turns up far beyond the cases anyone designed it for.

The Shape And Its Two Controls, $\mu$ And $\sigma$

A normal curve is a smooth, symmetric bell whose single highest point sits directly above the mean.
Because it is perfectly symmetric, the mean, median and mode all land at that same central value.
The mean $\mu$ sets the centre: Change $\mu$ and the whole bell slides left or right without changing shape.
The standard deviation $\sigma$ sets the width: A small $\sigma$ gives a tall narrow bell; a large $\sigma$ gives a short wide one, because the same total area is spread over more values.
We write $X \sim N(\mu,\, \sigma^{2})$, where the second entry is the variance $\sigma^{2}$, not the standard deviation.

Note

The total area under any normal curve is exactly 1, because the variable must take some value.
A probability is therefore an area under the curve, and the tails approach the axis without ever touching it.

The 68 95 99.7 Rule: A Quick Feel For The Spread

Because the shape is always the same once distance is measured in standard deviations, fixed percentages of the data fall within fixed numbers of $\sigma$ from the mean.
This lets you sanity-check any normal answer without a calculator.
1. About 68% of the data lies within $\mu \pm \sigma$.
2. About 95% lies within $\mu \pm 2\sigma$.
3. About 99.7% lies within $\mu \pm 3\sigma$.

Example

Adult male heights are roughly normal with mean 175 cm and standard deviation 7 cm.
So about 68% of men are between 168 cm and 182 cm, one standard deviation either side.
And about 95% are between 161 cm and 189 cm, two standard deviations either side.

Reading Probabilities As Areas

Every normal probability question is really an area question.
$P(X < x)$ is the area to the left of $x$, $P(X > x)$ is the area to the right, and $P(a < X < b)$ is the strip in between.
Because the curve is continuous, $P(X < x)$ and $P(X \le x)$ are equal, so the single point makes no difference.

Exam technique

The normal distribution has no by-hand formula in the formula booklet for these areas, so every probability comes from your GDC.
The booklet does list the standardized variable $z = \dfrac{x - \mu}{\sigma}$ under Topic 4, but you only need that in subtopic 4.12.
At this stage you always work directly from $\mu$ and $\sigma$, so keep the calculator in the driving seat.

Normal Calculations On The GDC: Two Commands

One command turns values into probabilities, and the other turns probabilities back into values.
Knowing which one a question wants is half the battle.

You are given	You want	GDC command
mean, SD and boundary values	a probability (area)	normalcdf
mean, SD and a probability	a boundary value	invNorm

For normalcdf you enter the lower bound, upper bound, mean and standard deviation, using a huge number for a missing one-sided bound:

Event	GDC entry
P(X < a)	normalcdf(-1E99, a, mean, SD)
P(X > a)	normalcdf(a, 1E99, mean, SD)
P(a < X < b)	normalcdf(a, b, mean, SD)

Common Mistake

The GDC wants the standard deviation $\sigma$, but the notation $N(\mu, \sigma^{2})$ gives the variance, so square-root it first if needed.
For "greater than" use the right tail, or take the complement $1 - P(X < a)$.

Example question

The mass $X$ of an apple is normally distributed with mean $\mu = 50$ g and standard deviation $\sigma = 10$ g. Find (a) $P(X < 60)$ and (b) $P(45 < X < 65)$.

Solution

a): area to the left of 60, so $P(X < 60) = \texttt{normalcdf}(-1\text{E}99, 60, 50, 10)$.
$= 0.841$ (3 s.f.).
(b): area of the strip, so $P(45 < X < 65) = \texttt{normalcdf}(45, 65, 50, 10)$.
$= 0.625$ (3 s.f.).

Inverse Normal: From An Area Back To A Value

Sometimes you know the probability and want the value that produces it, such as the 90th percentile or the mark that only the top 5% beat.
This is the inverse normal, and the key rule never changes: invNorm works from the area to the LEFT of the value.

Common Mistake

invNorm always expects the left-hand area, the probability of being below the value.
For "the top 10%", the value has 90% below it, so enter 0.90, not 0.10.
Sketch the curve and shade the area first, which stops almost every inverse-normal error.

Example question

Test scores are normally distributed with mean 100 and standard deviation 15. (a) Find the score below which 90% of candidates lie. (b) Find the score that only the top 5% exceed.

Solution

(a): 90% lies below, so the value is $\texttt{invNorm}(0.90, 100, 15)$.
$= 119$ (3 s.f.).
b): the top 5% exceed it, so 95% lie below it: $\texttt{invNorm}(0.95, 100, 15)$.
$= 125$ (3 s.f.).

See The Area Move As The Boundary Slides

The link between a boundary value and a probability is easiest to feel by watching the shaded area grow and shrink as the boundary moves across the curve.
The whole subtopic reduces to one idea: a normal probability is an area under the bell, and your GDC measures that area from $\mu$ and $\sigma$.

Active recall

In the notation $X \sim N(\mu, \sigma^{2})$, is the second number the standard deviation or the variance, and why does that matter on the GDC?
State the approximate percentage of data within one, two and three standard deviations of the mean.
Which GDC command turns boundary values into a probability, and which turns a probability into a boundary value?
To find the value that the top 20% of a normal distribution exceeds, what left-hand area do you enter into invNorm?
Explain why $P(X < x)$ and $P(X \le x)$ are equal for a normal distribution.

IB Mathematics Analysis and Approaches SL 4.9—normal Distribution and Calculations Notes

IB Mathematics Analysis and Approaches SL 4.9—normal Distribution and Calculations Notes

The Normal Distribution: Why So Many Things Cluster Around An Average

The Shape And Its Two Controls, $\mu$ And $\sigma$

The 68 95 99.7 Rule: A Quick Feel For The Spread

Reading Probabilities As Areas

Normal Calculations On The GDC: Two Commands

Inverse Normal: From An Area Back To A Value

See The Area Move As The Boundary Slides

The Normal Distribution: Why So Many Things Cluster Around An Average

The Shape And Its Two Controls, $\mu$ And $\sigma$

The 68 95 99.7 Rule: A Quick Feel For The Spread

Reading Probabilities As Areas

Normal Calculations On The GDC: Two Commands

Inverse Normal: From An Area Back To A Value

See The Area Move As The Boundary Slides

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

IB Mathematics Analysis and Approaches SL 4.9—normal Distribution and Calculations Notes

IB Mathematics Analysis and Approaches SL 4.9—normal Distribution and Calculations Notes

The Normal Distribution: Why So Many Things Cluster Around An Average

The Shape And Its Two Controls, $\mu$ And $\sigma$

The 68 95 99.7 Rule: A Quick Feel For The Spread

Reading Probabilities As Areas

Normal Calculations On The GDC: Two Commands

Inverse Normal: From An Area Back To A Value

See The Area Move As The Boundary Slides

The Normal Distribution: Why So Many Things Cluster Around An Average

The Shape And Its Two Controls, $\mu$ And $\sigma$

The 68 95 99.7 Rule: A Quick Feel For The Spread

Reading Probabilities As Areas

Normal Calculations On The GDC: Two Commands

Inverse Normal: From An Area Back To A Value

See The Area Move As The Boundary Slides