# Stats - Normal Distribution

> Source: https://ollybritton.com/notes/a-level/maths/topics/stats/normal-distribution/ · Updated: 2021-05-25 · Tags: school, stats, probability, distributions

## See Also
- [Stats - Binomial Distribution](https://ollybritton.com/notes/a-level/maths/topics/stats/binomial-distribution/)

## Flashcards
##### What is the notation for $X$ being a random variable following a normal distribution with mode $\mu$ and standard deviation $\sigma$??
$$
X ~ N(\mu, \sigma^2)
$$

##### ![PHOTO NORMAL DISTRIBUTION](normal-distribution.png) What is this??
The normal distribution.

##### What is the area under the normal distribution??
$$
1
$$

##### What is required for a random variable $X$ to follow a normal distribution??
It has to be continious.

##### How would you find $P(170 < X < 190)$ for the normal distribution??
Find the area under the curve between $170$ and $190$.

##### What proportion of data is within one standard deviation ($\pm\sigma$) of the mean in a normal distribution??
$$
0.68
$$

##### What proportion of data is within two standard deviations ($\pm2\sigma$) of the mean in a normal distribution??
$$
0.95
$$

##### What proportion of data is within three standard deviations ($\pm3\sigma$) of the mean in a normal distribution??
$$
0.997
$$

### 2021-06-10
##### $$P(X < a) = 0.1$$ How can you solve something like this??
Use the inverse normal distribution on the calculator.

##### $$P(16 < X < a) = 0.3$$ How can you rewrite something like this??
$$
P(X < a) = 0.3 + P(X < 16)
$$

##### $$P(X > a) = 0.7$$ How can you REWRITE something like this??
$$
P(X < a) = 1 - 0.7 = 0.3
$$

##### $$P(b < X < 16) = 0.4$$ How can you rewrite something like this??
$$
P(X < b) = P(X < 16) - 0.4
$$

### 2021-06-24
##### Why do we standardise normally-distrubted variables??
So that we can use standard results and values for probabilities.

##### What is the mean of the standard normal distribution??
$$
0
$$

##### What is the standard deviation of the standard normal distribution??
$$
1
$$

##### How would you write that $Z$ follows a standard normal distribution??
$$
Z \sim N(0, 1^2)
$$

##### How can you use the standard normal distribution for a random variable $X$??
Code the data so that it fits.

##### What is the coding for converting $X$ to a normally distributed $X$??
$$
Z = \frac{X - \mu}{\sigma}
$$

##### What's another way of writing $P(Z < a)$??
$$
\Phi(a)
$$

##### How could you rewrite $$P(Z > a) = 0.4$$??
$$
P(Z > a) = 1-0.4 = 0.6
$$

##### How could you rewrite $$P(0 < z < a) = 0.4$$??
$$
P(Z < a) = 0.5 + 0.4 = 0.9
$$

##### How could you rewrite $$P(-a < Z < a) = 0.4$$??
$$
P(-a < Z < 0) = \frac{0.4}{2} = 0.2 \\\\
P(-a < Z < 0) = 0.2 \\\\
P(Z < a) = 1 - 0.2 = 0.8
$$

### 2021-06-29
#### You know $$X \sim N(\mu, 3^2)$$ and $$P(x < 20) = 0.2$$ What is the process, but not the calculations, in order to find the value of $\mu$??
* Find the equivalent standardised $Z$ value such that $P(z < 20) = 0.8$.
* Undo the coding for $Z$ and solve for $\mu$.

#### $$P(X < 20) = 0.2$$ has been transformed into $$P(Z < 0.84162) = 0.8$$ for $X \sim N(\mu, 3^2)$. What's the next step??
$$
0.84162 = \frac{20 - \mu}{3}
$$

### 2021-07-08
##### What are the two conditions for approximating the binomial distribution $X \sim B(n, p)$ using the normal distribution??
* $n$ is large
* $p \approx 0.5$

##### $$X \sim B(n, p)$$ What is the value of $\mu$ for approximating the binomial distribution with the normal distribution??
$$
\mu = np
$$

##### $$X \sim B(n, p)$$ What is the value of $\sigma$ for approximating the binomial distribution with the normal distribution??
$$
\sigma = \sqrt{np(1 - p)}
$$

##### What is a continuity correction??
Approximating a discrete range using a continous one.

##### Why can't you use $P(Y = 1)$ instead of $P(X = 1)$ when approximating a binomially-distrubted $X$ with a normal distribution??
The normal distribution is continous whereas the binomial distribution is discrete.

##### What is the two-step process for doing continuity correction??
* If $>$ or $<$, convert to $\ge$ or $\le$
* Enlarge the range by $0.5$ at each end

### 2021-09-08
##### If you have 10 things you want to sample the mean of, and you sample them over and over again with a sample size of 4, what are you creating??
A distribution of sample means.

##### If a population $X$ is $$X \sim N(\mu, \sigma^2)$$ What is the sample distribution for repeatedly sampling that population with a size of $n$??
$$
\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)
$$

##### What is the formula for the variance of a sample distribution of size $n$??
$$
\frac{\sigma^2}{n}
$$

##### If you want to halve the standard deviation around the true mean in a sample distribution of size $n$, what factor do you need to increase the size of the sample by??
$$
\times 4
$$

##### What two distributions do you write down when doing a hypothesis test for the normal distribution??
* $X \sim N(\mu, \sigma^2)$
* $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$

### 2021-10-22
##### What's another name for the standard deviation of the distribution of sample means??
The standard error of the mean.

##### What is the "distribution of sample means" often abbreviated to??
The sampling distribution.

##### How would you code the data for $$\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$$ for $z$??
$$
z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}
$$

##### What are the hypotheses for a hypothesis test using a normal distribution??
Statements made about mean $\mu$.

##### If student test scores are normally distributed around a mean of $65$ with a standard deviation of $10$, how could you write the probability distribution for $X$??
$$
X \sim N(65, 10^2)
$$

##### If student test scores are normally distributed around a mean of $65$ with a standard deviation of $10$, how could you write the probability distribution for $\bar{X}$ with a sample size of $8$??
$$
\bar{X} \sim N(65, \frac{10^2}{8})
$$

##### What are you using the sampling distribution for in a normal distribution hypothesis test??
To see how likely the observed sample was.

##### How would you find the critical region for a 5% significance value??
Find the inverse normal of $0.95$.

##### What would you calculate to show that a sampled mean of $5$ from $\bar{X}$ was high enough to reject the null hypothesis??
$$
P(\bar{X} > 5)
$$

### 2022-03-01
##### When doing a continuity correction, how do you know whether to add or subtract $0.5$??
Consider which one of the two will increase the total values under consideration.

##### What do you always need to change $>$ or $<$ when doing a continuity correction??
$$
\ge, \le
$$

### 2022-05-16
##### How would you find $$P(T < 2 | T > 0)$$??
$$
\frac{P(0 < T < 2)}{P(T > 0)}
$$

##### What is true about the median at value $x$ of a random variable $X$??
$$
P(X > x) = 0.5
$$

##### $$X ~ N(\mu, \sigma^2)$$ What would you "solve" to find the median value of a new distribution where $X$ cannot be less than $0$??
$$
\frac{P(X > t)}{P(X > 0)} = 0.5
$$

### 2022-05-17
##### What technique do I forget a lot when answering questions about modelling with a distribution??
Using conditional probability, A given B

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt