## What are the topics in statistics?

### Probability

In probability, we explore the idea of independent and dependent events. You will learn about calculating the chance that an event will occur using various methods such as tree and Venn diagrams and conditional and mutually exclusive events.

With Venn diagrams, you can figure out how events can happen at the same time.

Draw a Venn diagram for the following data:

U = numbers under 20

A = even numbers

B = Multiples of 3

Solutions:

Venn diagram example

Calculate \(P(A \cap B)\)

Solutions:

There are 19 numbers in the entire set.

There are 3 numbers inside the intersection.

So \(P(A \cap B) = \frac{3}{19}\)

### Collecting data

Here, we look at sampling, including different methods of sampling and the different types of data. Some people find sampling questions the easiest to answer in an exam, but they can also be quite wordy, so it is important to pay attention – and then you will understand precisely what you are being asked to do.

There are a few different ways to categorize data. We can categorize data as quantitative or qualitative, as well as descriptive and inferential.

There are 400 students in the school, 250 girls and 150 boys. Explain how to take a stratified sample of 40 students in the school.

Solution.

We want to pick 25 girls and 15 boys (so that we have the same proportion as the entire population). One method is to assign all the girls a number from 1-250. Then using a random number generator, generate 25 numbers and then pick those 25 girls.

Repeat the same for the boys: assign them a number from 1-150. Then use your random number generator to generate 15 numbers and then pick those 15 boys.

### Measures of location and spread

We need to analyze the data we collect, and the best way to do this is by using measures of location and spread. This enables us to compare data using the following

- The mean and standard deviation.
- Using simple data analysis such as the mode and the range.
- Finding quartiles and percentiles.
- Using algebra to code this data.

On a randomly chosen day, each of the 32 students in a class recorded the time (t) in minutes to the nearest minute that it took them to get to school. Find the mean and standard deviation from the following data:

\[\sum t = 1414 \text{ and } \sum t^2 = 69378\]

Solutions:

Mean: \(\frac{\sum t}{n} = \frac{1414}{32} = 44.1875\)

Default Deviation: \(\sqrt{\frac{\sum t^2}{n} - \Big( \frac{\sum t}{n} \Big)^2} = \frac{69378}{32} - (44.1875)^2 = 215.52734375\)

### Statistical distributions

A vital part of statistics is understanding the distribution of data. Distributions are essentially mathematical functions that give the probability that a function will occur. We will look at two main distributions, binomial distribution and normal distribution.

Binomial distribution applies whenever there are two mutually exclusive possible outcomes of an experiment. If an experiment with the probability of the outcome happening being p is performed n times, the probability of this outcome happening n times is:

\(P(X = a) = \left( \begin{array} {c} n \\ a \end{array} \right) p^{a} (1-p)^{n-a}\) with \(\left( \begin{array} {c} n \\ a \end{array} \right) = \frac{n!}{a!(n-a)!}\) (also written as \(^n{C}_a\))A die is tossed 10 times. The outcome of rolling 5 exhibits a binomial distribution: \(X \sim B(10, \frac{1}{6})\). Calculate \(P(X \leq 3)\).

Solution.

This is as simple as calculating P(X = 0), P(X = 1), P(X = 2) and P(X = 3) and summing them all together. Therefore: \(P(X = 0) = \left( \begin{array} {c} 10 \\ 0 \end{array} \right) (\frac{1}{6})^0 (\frac{5}{6})^{10} = 0.1615055829; \quad P(X=1) = \left( \begin{array} {c} 10 \\ 1 \end{array} \right) (\frac{1}{6})^1 (\frac{5}{6})^9 = 0.3230111658; \quad P(X=2) = \left( \begin{array} {c} 10 \\ 2 \end{array} \right) (\frac{1}{6})^2 (\frac{5}{6})^8 = 0.2907100492; \quad P(X=3) = \left( \begin{array} {c} 10 \\ 3 \end{array} \right) (\frac{1}{6})^3 (\frac{5}{6})^7 = 0.155043596\)

Summing all of these together we get: \(P(X \leq 3) = 0.9302721575\)

In some exams, you will have access to a formula booklet with a dedicated section for statistics. Check your exam board's website for this. The most useful formulas are the ones for binomial distribution and mutualistic probability; however, the booklet will also contain statistical tables. You might not need these if you can use a calculator, but they show probability values at significance levels on distributions.

## Hypothesis testing

Hypothesis testing involves using distribution to calculate whether or not we can say a statement ( a hypothesis) is true. In the hypothesis testing topic, we will look at conducting one-tailed and two-tailed tests and stating a null hypothesis.

A coffee shop claims that a quarter of the cakes sent to them are missing cherries on top. To test this claim, the number of cakes without cherries in a random sample of 40 is recorded. Using a 5% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a missing cherry is 0.26.

Solutions:

This is a two-tailed test, meaning we will need to look at both ends. Start at a random number on both ends.

Lower end:

P(X ≤ 6) = 0.07452108246

P(X ≤ 5) = 0.03207217407

P(X ≤ 4) = 0.01136083855

As this is a two-tailed test we want to be as close to 0.025 as possible:

P(X ≤ 4) < 0.025 < P(X ≤ 5).

Upper end:

P(X ≥ 16) = 1 - P(X ≤ 15) = 0.03703627013

P(X ≥ 17) = 1 - P(X ≤ 16) = 0.0171086868649

Again we want to be as close to 0.025 as possible so:

P(X ≥ 17) < 0.025 < P(X ≥ 16)

Our critical regions are therefore P(X ≤ 4) and P(X ≥ 17).

### Representing data

In the representing data topic, we will look at graphical methods to showcase data. These include histograms, box plots and cumulative frequency. We will also look at ways that we can find outliers in data, and how to deal with data anomalies.

If you are studying for A-level exams, some exam boards prove a large data set eg a spreadsheet containing weather data from airports in the UK and around the world. You don't have to memorize any data but what you do have to do is familiarize yourself with the different types of data it contains, and the units of these data.

## Statistics - Key takeaways

- Statistics can be broken up into many concepts.
- A lot of the concepts from Pure Mathematics are used in Statistics such as Binomial Expansion in Binomial Distribution.
- The Mathematical process in Statistics is the same as that in Pure Mathematics.

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##### Frequently Asked Questions about Statistics

What is statistics?

Statistics is the branch of Mathematics that deals with collecting and analysing numerical data in large quantities.

What are the three types of statistics?

The three types are descriptive, inferential and quantitative.

What are descriptive statistics?

Statistical measures to organise, present and summarise data in an informative manner.

What does variance mean?

Variance is a measure of spread equal to the standard deviation squared.

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