#### Qualitative and Quantitative Statistics

Statistical data comes in one ogf two broad types Qualitative and Quantitative. Qualitative data is descriptive – a colour, type or one of a category. The categories that describe qualitative data may be ordered or not. We may order colours in order of...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1062-qualitative-and-quantitative-statistics.html#### Calculating Regression Lines

The correlation coefficient can be found if necessary to found how good a relationship we have. You may be given summary statistics etc but I will illustrate an example from scratch. Example: Find the equation of the regression line of height above...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1064-regression-lines.html#### Simultaneous Equations Problems Involving the Normal Distribution

set of data and so cannot find the mean and standard deviation from a set of readings. However, given JUST TWO suitable statistics we can find the mean and standard deviation of a list. This is because there are two constants we must find, and any set...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1065-simultaneous-equations-problems-involving-the-normal-distribution.html#### The Correlatioin Coefficient

table. x 2 4 7 3 5 1 y 10 5 2 9 4 10 To find the correlation coefficient it is most efficient to first find the summary statistics. Your calculator may have functions to help. They are The correlation coefficient is then calculated from r= This means...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1045-the-correlatioin-coefficient.html#### Mode, Median and Mean

While the mode, median and mean are all measures of average, there are fundamental differences between them all: when you hear politicians splitting hairs, they are often splitting hairs over the difference between the average – mean – value, or the...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1054-mode-median-and-mean.html#### Introducing the Normal Distribution

The normal distribution is the most useful, adaptable, flexible, used statistical distribution there is. All other distributions are, in a sense, subsumed by it because of the Central Limit Theorem, which says that in some circumstances any statistical...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1052-introduction-to-the-normal-distribution.html#### More Calculations Involving The Normal Distribution - Finding Probabilities

Suppose we have a distribution We will find using the diagram on the left. This corresponds to a probability of 0.6915 hence Now we will find illustrated by the diagram on the left. Ignoring the minus sign we go to the normal distribution tables and...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1055-more-calculations-involving-the-normal-distribution-finding-probabilities.html#### Mutually Exclusive, Exhaustive, Independent Events Illustrated on Venn Diagrams

Mutually exclusive events are events that cannot both happen at the same time: You either like bovril or hate it. You either do what we say or you die! (George Bush's sentiment). We can illustrate mutually exclusive events A and B on a Venn diagram....

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1056-mutually-exclusive-exhaustive-independent-events-illustrated-on-venn-diagrams.html#### How to Deal With Negative z Values When Using the Normal Distribution

The z transform for the normal distribution is written Thenormal distribution tables are used as below: a)Find a value of usinga given value of usingthe above formula, then going to the normal distribution tables tolook up a probability using this...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1057-how-to-deals-with-negative-z-values-when-using-the-normal-distribution.html#### Pascal's Triangle

Pascal's triangle provides a quick and easy way to calculate the coefficients of the binomial expansion, The entries in the table are calculated by adding the entties in each table immediately above to the left and right, with the exceptions of the...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1058-pascal-s-triangle.html#### Populations, Samples and Censuses

A population is the set of individuals in a group. Often information needs to be collected about the population, Taking a census and taking a sample are methods of collecting data about the population. A govenrnent for example needs to plan the supply...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1059-populations-samples-and-censuses.html#### Probability Mass Functions - Discrete Distributions

Every probability distribution has a probability density function, or probability mass function in the case of a discrete distribution, in terms of which it is usually defined. The probability density function for the normal distribution is for the...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1060-probability-density-functions-discrete-distributions.html#### Probability Tables

We start with a raw table of values and for each value, a probability of getting that value. x 1 2 3 4 5 P(X=x) 0.1 0.3 0.15 0.25 0.2 From this table we can find There are formulae for and They are Suppose we need to find and No matter how we transform...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1061-probability-tables.html#### Quartiles & Median From Frequency Tables

We start with the table below which summarises the daily sales in a newspaper shop over a year for the 360 days the shop was open. Sales Number of Days 1 -200 166 201- 400 100 401 - 700 59 701 - 1000 30 1001 -1500 3 To find the lower quartile we use...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1063-quartiles-and-median-from-frequency-tables.html#### Skewness

The normal distribution is symmetrical – small values are as likely as large values. The probability distribution for the normal distribution is symmetrical about the mean. This is not the case for most distributions however. Most probability...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1066-skewness.html#### Space Diagrams

A space diagram is a table which displays all the possible outcomes. They are usually used for discrete distributions and are especially useful when 2 or more trials are carried out simultaneously and the results must be combined in some way. For...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1067-space-diagrams.html#### Standard, Deviation, Variance and Bias

Given a flock of sheep, we can measure the masses of a sample of sheep and calculate the mean mass and the standard deviation of the masses of the sheep in the sample. Suppose the masses of the sheep are 50, 55, 52, 60, 54, 49, 59, 58 The sample mean...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1068-standard-deviation-variance-and-bias.html#### Summary of Formulae - Conditional Probability, Independent Events etc

This is only true if A and B are independent. You can never assume A and B are independent. You must either know it or prove it. This is always true for two events A and B. The left hand side P(A divides B) is the probability that A will happen given...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1069-summary-of-formulae-conditional-probability-independent-events-etc.html#### The Binomial Distribution

The name “The Binomial Distribution” is derived from the binomial expansion, because if you use the formula for binomial expansion, you can substitute and and then the probability of obtaining successes in attempts is given by the th term There are...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1070-the-binomial-distribution.html#### The Poisson Distribution

When events occur at a constant average rate, the distribution decribing the events can be modelled by a Poisson distrubtion. Examples include: Plane crashes Spelling or grammatical errors in a textbook School shootings (only in America) It is...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1071-the-poisson-distribution.html#### The Standardized Score

The standardized score is a statistical tool thatallows comparison of related quantities that are normallydistributed. A very good use is test scores. When a student wishes tocompare their own results with everybody elses, they may stanaridizetheir...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1072-the-standardized-score.html#### Venn Diagram For Three Independent Events

If three events A, B and C are mutually independent, then Suppose that and then The Venn diagram starts as This includes the probability so subtract this to obtain the diagram below. Similarly for A and C and B and c respectively. Subtract to get 0.036...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1073-venn-diagram-for-three-independent-events.html#### Venn Diagrams 1

Venn diagrams are a means to display categories of data graphically. They are more flexible than contingency tables, allowing complex reasoning, and have many applications in set theory. Typically each set is illustrated by a bubble, allowing...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1074-venn-diagrams-1.html#### Venn Diagrams 2

Calculations involving Venn diagrams use the formulae: and if the events and are independent, where for example, P(A) indicates the probability of A happening. The best way to illustrate the use of these examples is by way of examples. A Venn diagram...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1075-venn-diagrams-2.html#### Mean and Variance of the Poisson Distribution

The Poisson distribution models events occurring at a constant average rate so that in each fixed time period we expect events to occur. The probability that events will occur in a single fixed time period is and the variance of the number of events in...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1053-mean-and-variance-of-the-poisson-distribution.html#### Independent Events

Tow events A and B are statisitically independent if neither influences the oother. The probability of event A occurring does not influence the probability of event B occurring and vice verse. Suppose neither event A or event B has occurred yet. Event...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1051-independent-events.html#### Drawing Histograms

A histogram is a type bar chart using intervals on the x -axis, with the modification that the interval sizes are not fixed. The frequencies are defined by the areas of the bars, and the y – axis is not a frequency axis, but a frequency density axis. A...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1050-histograms.html#### Finding the Mean From Frequency Tables

To find the mean of a list of numbers we add the numbers up and divide by how many numbers there are. For example the mean 1, 4, 6, 6, 7 is However sometimes we have a list in the form of a table. The table below is a fictitious summary of the number...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1049-finding-the-mean-from-a-frequency-table.html#### Finding the Quartiles and Median of a List of Numbers

To find the median and quartiles of a list of numbers, first sort them – if they are not already sorted. If there are n numbers in the list: The median will be the th number . If this is not a data point – is even – find the averare of the th and th...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1048-finding-the-quartiles-and-median-of-a-list-of-numbers.html#### Discrete and Continuous Random Variables

A random variable represents the possible outcomes we could observe on measuring some quantity. Ransdom variables can be either discrete or contionuous. A discrete random variable has possible values The number of possible values that can take might be...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1047-discrete-and-continuous-random-variables.html#### Cumulative Frequency Curves

Length ( x mm) Frequency 2 4 8 14 6 4 2 We find the cumulative frequencies by adding up the frequencies as we go down the table. Length ( x mm) Cumulative Frequency 2 6 14 28 34 38 40 Cumulative frequency curves are useful statistical tools that...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1046-cumulative-frequency-curves.html#### Continuous Probability Distributions

If a random variable has a continuous probability distribution, then may take any value in an interval. The end points of the interval may be included or not. If denotes possible values of the random variable then means ' may take values greater than...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1044-continuous-probability-distributions.html#### Conditional Probability

Conditional probability questions arise when two possibly linked events, and occur. We ask, given that has happened, what is the probability that will happen? The equation we use is, the meaning of which is illustrated in the diagram below. What we are...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1043-conditional-probability.html#### Calculations Involving The Normal Distribution - Finding Probabilities

If you look at the tables for the normal distribution, along the top and left borders there are z – values. If we have a Normal distribution and we want to find then we can use the – transform to find a – value then find the probability for this –...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1042-calculations-involving-the-normal-distribution-finding-probabilities.html#### Boxplots or Box and Whisker Diagrams

A box-and-whisker plot or boxplot is a visual display of some of the five figure summary of a data set. It shows the minimum value, the lower quartile the median the upper quartile and the maximum value For the data set 2, 4, 4, 6, 7, 8, 12, 13, 17,...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1041-boxplots-or-box-and-whisker-diagrams.html#### Bayes' Theorem

Suppose we observe a random variable and wish to make inferences about another random variable where is drawn from some distribution From the definition of conditional probability, Again from the definition of conditional probability, we can express...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1040-bayes-theorem.html#### Advanced Questions - Venn Diagrams

More complicated question on Venn diagrams involve the use of simultaneous equations, or the introduction of a dummy variable x. Example Sketch the Venn diagram. (1) (2) Divide (1) by (2) to get Substitute these into the equation Then and from (1) We...

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The cathode ray tube is a very common device used in television sets and computer screens. Inside the tube is a vacuum, so the electrons are free to move and do not hit any gas molecules. Electrons are emitted from a negative cathode at A, which is...

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https://astarmathsandphysics.com/#### The Cases For and Against Nuclear Power

Before nuclear power there was the nuclear bomb, and the presence of the nuclear bomb casts a shadow over the debate about nuclear power, even between countries. Many countries think that nuclear bombs are being developed under the cover of nuclear...

https://astarmathsandphysics.com/gcse-physics-notes/833-the-cases-for-and-against-nuclear-power.html#### Rutherford's Model of the Atom

An experiment which involved firing alpha particles at a piece of thin gold filed lead Ernest Rutherford to draw up a new model of the atom as a very small heavy nucleus surrounded by orbiting electrons. To the great surprise of Rutherford, and the two...

https://astarmathsandphysics.com/ib-physics-notes/the-history-and-development-of-physics/1322-rutherford-s-model-of-the-atom.html#### Graphical Representation of Uncertainty

In many situations the best method to present an assess data is to use a graph. Error bars enable goodness of fit to a specific statistical model to be assessed. Since the error represents the uncertainty range, the best fit curve should pass through...

https://astarmathsandphysics.com/ib-physics-notes/measurements-units-uncertainty-and-principles/1342-graphical-representation-of-uncertainty.html#### Error Anaysis 2

Error has to do with uncertainty in measurements that nothing canbe done about. A metre rule might measure to the nearest mm. If ameasurement is repeated, the length obtained may differ and none ofthe measurements can be preferred. Although it is not...

https://astarmathsandphysics.com/university-physics-notes/experimental-physics/1560-error-analysis-2.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...

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Gathering statistics involves time and effort. It is a necessary procedure – 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, but...

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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...

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