#### Error Analysis 2

Error has to do with uncertainty in measurements that nothing can be done about. A metre rule might measure to the nearest mm. If a measurement is repeated, the length obtained may differ and none of the measurements can be preferred. Although it is...

https://astarmathsandphysics.com/a-level-physics-notes/principles-dimensions-units-and-error-analysis/2978-error-analysis-2.html#### Maxwell's Demon

James Clerk Maxwell introduced a now-famous thought experiment that highlighted the contrast between the statistical nature of entropy and the deterministic nature of the underlying physical processes. This experiment, known as Maxwell's demon,...

https://astarmathsandphysics.com/a-level-physics-notes/thermal-physics-and-gases/3048-maxwell-s-demon.html#### The Recurrence Paradox

The second law of thermodynamics states that energy flows from hotter to colder bodies in a way that increases the total disorder of the Universe, or equivalently, that the particles of the Universe tend to move to states that are 'more probable'. In...

https://astarmathsandphysics.com/a-level-physics-notes/thermal-physics-and-gases/3069-the-recurrence-paradox.html#### An Introduction to Hypothesis Testing

Hypothesis testing is used to test whether the value of a mean or standard deviation has changed over time, whether the mean or standard deviation of two samples are equal, whether dogs prefer Dogslop or Kanineswill etc. You are testing always a null...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3688-an-introduction-to-hypothesis-testing.html#### Confidence Intervals

There are various ways we can find out how extreme an event is. We can find the interquartile range and say whether or not the event is an outlier. We can apply the z transform if it is a normal distribution and find the probability of events more...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3699-confidence-intervals.html#### Hypothesis Testing With the Binomial Distribution

If a manufacturer claims superiority for any of their products or a great deal rests the proportion of components that exceed a certain lifetime, then that claim or proportion probably needs to be tested for legitimacy or accuracy. We can test a wide...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3711-hypothesis-testing-with-the-binomial-distribution.html#### Outliers

An outlier of a statistical sample is an extreme value – the very rich man, the dwarf, the Spruce Goose – a large plane built by reclusive billionaire Howard Hughes. It flew only once and now sits in a museum – a monument to people with too much money....

https://astarmathsandphysics.com/a-level-maths-notes/s1/3721-outliers.html#### Residuals

Suppose we have a set of data points relating one quantity to another. We can plot the points on a scattergraph and find the equation of the regression line of on so that is expressed linearly in terms of If we take one of the points from which the...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3731-residuals.html#### The Central Limit Theorem

Suppose we have a list of measurements of some quantity from some population. We don't know the underlying distribution, but for some purposes this does not matter. If we are only interested in analysing the mean of the list, there is a very useful...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3740-the-central-limit-theorem.html#### The Standardised Score

The standardized score is a statistical tool that allows comparison of related quantities that are normally distributed. A very good use is test scores. When a student wishes to compare their own results with everybody elses, they may stanaridize their...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3742-the-standardised-score.html#### Prediction and Extrapolation

Extrapolationis a bad idea. It involves using a set of data to make predictionsfor data outside the range used to construct a statistical model.Suppose for example that a drug is being tested. The drug is designedto lower cholesterol. The drug is...

https://astarmathsandphysics.com/a-level-maths-notes/s2/3763-prediction-and-extrapolation.html#### Type I and Type Errors - The Power of a Test

Conducting a hypothesis test is by nature an uncertainbusiness. Just be cause you do not reject the null hypothesis, itdoes not mean that the null hypothesis is true. There is in factinsufficient evidence from the hypothesis test to mean that you...

https://astarmathsandphysics.com/a-level-maths-notes/s2/3774-type-i-and-type-errors-the-power-of-a-test.html#### Biased Estimators

An estimator fora statistical parameter issaid to be biased if Bias is often impossible toavoid in practice and must be taken into account when statisicalcalculations are performed. Example: To estimate thenumber, ofnesting birds, scientists catch 100,...

https://astarmathsandphysics.com/a-level-maths-notes/s3/3777-biased-estimators.html#### Estimating Population Mean and Variance From a Sample

In practice most of statisitics involves taking and analysing samples. The most important statistical measures for any sample are the mean and variance (or the standard deviation ). We usually do not know the mean and standard deviation of the...

https://astarmathsandphysics.com/a-level-maths-notes/s3/3782-estimating-population-mean-and-variance-from-a-sample.html#### Unbiased Estimators

An estimator fora statistical parameter issaid to be unbiased if The mean of a sample isan unbiased estimator of the population mean Thiscan be used in a less obvious way in the following example. A bag has 500 beans of twocolours, black and white. A...

https://astarmathsandphysics.com/a-level-maths-notes/s3/3796-unbiased-estimators.html#### The Two Sample Paired t -test

The paired t test istypically used to analyse whether there is a before and afterdifference. It is the same population before and after, and thedifference for each individual is measured and a mean differencefound and analysed for significance. For...

https://astarmathsandphysics.com/a-level-maths-notes/s4/3818-the-two-sample-paired-t-test.html#### The Mann Whitney U Test

The Mann Whitney U test allows the comparison of two independent random samples (1 and 2). The Mann Whitney U statistic is defined as: - where samples of size and are pooled and R i are the ranks of sample 1. U can be resolved as the number of times...

https://astarmathsandphysics.com/a-level-maths-notes/s6/3821-the-mann-whitney-u-test.html#### Pie Charts

60 children go to school. There are various methods they use – they may walk, go by car, bus or train. The statistics are in the following table. Method of Going to School Frequency Angle Walk 21 Car 15 Bus 20 Train 4 Total 60 360 Because it is for a...

https://astarmathsandphysics.com/gcse-maths-notes/618-pie-charts.html#### Sampling Methods

Gathering statistics involves time and effeort. It is a necessary procecdure – governments need to know how many schools to build, or policemen to employ, and companies need to know which products people will buy. They could go out and ask everyone,...

https://astarmathsandphysics.com/gcse-maths-notes/645-sampling-methods.html#### Counting Permutations

Permutations is that part of statistics involving arrangements of objects, some of which fall in the same group and some of which fall in different groups. Each object is distinct from all the other objects, so we can tell each one apart, and if two...

https://astarmathsandphysics.com/ib-maths-notes/counting-permutations-and-combinations/962-counting-permutations.html