# Statistics (S1) - How Tos

How To revision cards on S1 Statistics including formulae, step-by-step instructions and examples.

- Mathematics
- Statistics, averages and distributionsNumerical methodsNumerical MeasuresProofProbabilityDistributionsLawsSymbolsMeasures of Spread
- AS
- AQA

- Created by: Betsy_2018
- Created on: 26-12-16 10:23

## Population and Sample Standard Deviation

**Population**

**σ = | ∑(x - µ)^2**

**o.o**

**√**

**n**- Use the population stadard deviation if it specifies that it is the entire population of data

- Use the population standard deviation if all data is present (e.g number of passengers on a us each day of the week: 102, 353, 401, 485, 209, 314, 175)

**Sample**

s = ** | ∑(x - µ)^2**

**..**.

**√ n-1**

- Use the sample standard deviation when is specifies that it is a sample

- estimates the population based on a sample

## Probability: Union and Intersection

- Event = set of outcomes
- Sample space = set of all possible outcomes
- Exhaustive events = set of outcomes cover all possible outcomes of the sample space
- Mutually exclusive events = 2 events can not occur at the same time
- Independent events = occurance of events do not affect others
- Complements = An event and the opposite of that event (e.g A and A')
- Conditional probability =
*given*= [(e.g B|A) = independent if P(A) = P(A|B)]

**Union** = **∪**= 'or', 'both'

*Addition Law = P(A∪ B) = P(A) + P(B) - P(A∩B)**Mutually Exclusive Law = P(AUB) = P(A) + P(B)*

**Intersection** = **∩** = 'overlap', 'and'

*Multiplication Law = P(A∩B) = P(A) x P(B|A)**Independent Law = P(A**∩B) = P(A) x P(B)*

## Standard Deviation Summary

- A much more precise
*measure of spread*than just the range or the interquartile range. - For S1, there are 5 (unfortunately!) formulae that you need to be able to recognise - they will be on your formula sheet - and are each for different types of data
- The standard deviation is the difference between each piece of data and the mean

**Symbols**

**σ** = Sigma = population standard deviation

**∑x **= total of data

**x̄** = sample mean

**µ** = population mean

**n** = number of pieces of data = **∑ƒ**

**s** = sample standard deviation

## Binomial Formula and Cumulative Binomial Tables

P(X = x) = (n r)P^{x}(1-p)^{(n-x)} = n! ÷ (n-X)!X!

**Example**

20 friends, probability that 5 will go swimming next week? p= 1/7 n= 20 r= 5

(20 5) 1/7^{5} x 6/7^{15} = 0.0914

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Cumulative Binomial Tables**

- This is the same concept, but instead of =, it is going to be < > ≤ ≥
- you will need to use the formula above, but add each term's applied formula together

**Example**

p= 0.25 n= 8 X~B(8, 0.25)

p(X<3) = P(X=0) + P(X=1) + P(X=2)

## Binomial Distribution Summary

**X** = successes = variable = 'pass or fail'

**n** = number of trials (number of things) taken **r** at a time = fixed

**P** = probability of successes

**Q** = probability of failure (1-**P**)

**X ~ B(n, p)** = binomial probability *Bi**= 2 outcomes*

*(n r) = nCr** *

*Note: the formula can be used OR the statistics table, each labelled n=[ ] Either way, draw a numberline! No graphs are drawn.*

*When using the tables, you may be required to do '1-P(X)' if you are trying to find (P≥?)*

*or P(X) - P(X-1) when trying to find an exact value*

*or P(X) - P(Y) to find a range*

## Mean and Variance of Binomial Distribution

**Mean**

= µ

= np

= (number of trials x probability)

**Variance**

= σ2

= np(1-p)

= (number of trials x probability)(1 - probability)

## Standardising a Normal Variable

z = x - µ / σ

x = µ + zσ

**Example**

mean = 50

x = 51

σ^2 = 25 (5^2)

Therefore, 51 - 50 / 5 = 1/5

P(Z<0.2) = 0.579 (3sf)

## Normal Distribution Types (1)

**Finding a probability using table 3, given x**

z = x-µ / σ

**Finding an x value using table 4, given a probability**

x = µ + zσ

Make x the opposite (e.g if you need to find the first 5%, switch to 95%) because p cannot <0.5

**Setting up 2 equations with unknown µ and σ, solving simultaneously**

Given 2 x values, x = µ + zσ, probability between the 2 x values

## Normal Distribution Types (2)

**Standard deviations away from the mean (Modelling Normal Distribution)**

**68%**area lies in the ranges of µ ± σ (1 standard deviation)**95.5%**area lies in the ranges of µ ± 2σ (2 standard deviations)**99.7%**area lies in the ranges of µ ± 3σ (3 standard deviatoons)

**Finding the sample mean**

sample size n

x̅ distributed by *µ* and standard deviation *σ/√n*

z = x - x̅ / *σ/√n*

*x = *x̅ + z(*σ/√n)*

## Normal Distribution Summary

- bell-shaped
- draw graphs!
- median = mode = mean = symmetrical
- total area = 1
- high population density close to the mean
- X ~ N(0, 1^2)

X is normally distributed with µ (mean) and σ2 (variance)

**Standard Normal Distribution**

Mean = µ = 0

Standard Deviation = 1

- Z ~ N(0, 1)2

Z is normally distributed with µ(=0) and σ2(=1)

## Normal Distribution Tables

For this, we are using Table 3 and Table 4.

Table 3 is to find the probability that Z (mean = 0 and variance = 1) is normally distributed to euqal less than or equal to z

**Example**

1.36 on a graph. P(Z≤1.36) = 0.913 on Table 3

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Table 4 is to find the value of z that satisfies P(Z≤z) = p

**Example**

p = 0.9

P(Z≤z) = 0.9

z = 1.2816 on Table 4

## The Critical Limit Theorem

x̅ ~N(µ, (σ/√n)^2)

z = x̅- µ / (σ/√n)^2

x̅ = µ + z(σ/√n)

**Example**

Weight of pebbles are distributed with mean 48.6g and standard deviation 8.5g. Random sample of 50.

P(x̅<49.0g) n= 50

z = 49.0 - 48.6 / 8.5/√50 = 0.33 (2dp)

P(x̅<49.0g) = P(Z<0.33) = 0.629 (3sf)

## Related discussions on The Student Room

- I don't understand statistics »
- Maths S2 or D1? - »
- Do you need to know S1 to understand S2? »
- S2 - How to tell if something is/isn't a statistic? »
- M1 or S2? »
- S1 exam paper help! »
- Statistics or Mechanics? »
- AQA A Level Statistics »
- Additional Maths question help Cosine formula »
- Self-teaching S2 Maths »

## Comments

No comments have yet been made