Statistics

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Handling data

Population and sampling:

Population = Whole set of items that are of interest

Sampling units = Individual units of a population

Sampling frame = Sampling units of a population individually named or numbered to form a list

Census = Observes or measures every member of a population

Sample = Selection of observations taken from a subset of the population to be used to find out information about the population as a whole

Census:

Advantages: Completely accurate result

Disadvantages: Time consuming, expensive, hard to process large quantity of data, cannot be used when testing destroys the data

Sample:

Advantages : Less time consuming and expensive than a census, less data to process, fewer people need to respond

Disadvantages: May not be accurate, sample may not be large enough to give information about sub-groups of the population

Random sampling techniques:

Simple random sampling:

Advantages: Free of bias, easy and cheap for small populations and small samples, each sampling unit has a known and equal chance of selection

Disadvantages: Sampling frame is needed, not suitable when the population size or sample size is large

Stratified sampling:

Advantages: Accurately reflects the population structure, guarantees proportional representation of groups within a population

Disadvantages: Population must be classified into distinct strata, section within each stratum suffers from the same disadvantages as simple random sampling

Systematic sampling:

Advantages: Simple and quick to use, suitable for large samples and large populations

Disadvantages: Sampling frame is needed, can introduce bias if the sampling frame is not random (alphabetical list, same surname for example)

Non- random sampling techniques:

Opportunity sampling:

Takes samples from members of the population you have access to until you have a sample of the desired size

Advantages: Easy to carry out, inexpensive

Disadvantages: Unlikely to provide a representative sample, highly dependant on the individual researcher

Quota sampling:

When you know you want distinct groups to be represented in your sample, decide how many members of each group you wish to sample in advance and use opportunity sampling until you have a large enough sample for each group

Advantages: No sampling frame required, quick, easy, inexpensive, allows a small sample to still be representative of the population, allows for easy comparison between different groups within a population

Disadvantages: Can introduce bias, population must be divided into groups which can be inaccurate or costly, non-response are recorded as such, increasing scope of study increases number of groups which adds time and expense

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Simple random sampling:

Calculator: 1 > RUN MATHS > OPTN > PROB > RAND > Int > ( 00, __)

Never Swim In Cold Soup

Number list from 00 to ___

Select __ random digits/numbers

Ignore repeats

Continue until you have __ numbers

Select corresponding items from the data set

Stratified sampling:

Total group/ Total population available x sample size

Systematic sampling:

Members of the population are chosen at regular intervals

Interval length = Amount of items in population / Sample size

Find the interval length

Do a simple random sample using the interval length to find the first random number

Using this number count up in regular intervals using the interval length

Select the corresponding numbers until you reach your sample size

Measures of central tendency and measures of spread:

Measures of average:

Mode: Used when data is qualitative, or quantitative with either a single mode or 2 (bimodal). Not very informative if each value only occurs once

Median: Used for quantitative data when there are extreme values as they do not affect the median

Mean: Used for quantitative data and uses all the pieces of data, gives a true measure the data. However can be affected by extreme values

Measures of spread:

Range = Difference between highest and lowest values

Interquartile range = Difference between upper and lower quartile (Q3 - Q1)

Standard deviation ( sigma standard deviation on calculator) > Best to use

Variance = Measures of the average deviation of the values from their mean (SD squared)

Linear interpolation:

USE A TRIANGLE

Summation statistics:

Formulas are given in the front of the booklet.

SD squared = Variance

Coding:

y = x - a/b > x= by + a

sigma y = sigma x / b > sigma x = b x sigma y

Data Set:

Daily mean cloud cover measured in oktas (eighths of the sky covered by cloud)

Daily mean visibility measured in decametres (Dm)

Daily mean pressure measured in hectopascals (hPa)

tr = trace which is between 0 and 0.05 mm of rain, we treat trace amounts as 0 in numerical calculations as anything less than 0.005 would be 0.0 to 1d.p.

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Representation of data

Cumulative frequency:

Use the upper-bound and frequencies are added hence the name "cumulative" frequency

Frequency polygons:

Use the midpoint and frequency

Join the points with straight lines

Histograms:

Area of the bar is proportional to the frequency in each class

Area of bar = K x frequency

A= kf

If k = 1 then frequency density = frequency / class width

Joining the middle of the top of each bar on a histogram forms a frequency polygon

Correlation and Regression:

Scatter diagram: x axis = independent/ explanatory variable , y axis = dependant/ response variable

No correlation, positive, negative, strong, weak, no linear correlation( parabola for example)

Casual relationship = If a change in one variable causes a change in the other, just because two variables show correlation doesn't necessarily mean that they have a casual relationship

When two variables are correlated you need to consider the context of the question and use your common sense to determine whether they have a casual relationship

Regression: Equation for the linear relationship between 2 variables x and y, often called line of best fit, general equation: y = a + bc (y = + mc) where b denotes the gradient and a denotes the y-intercept

If you know a value for the independent variable from a bivariate data set you can use the regression line to make an estimate for the corresponding value of the dependent variable > doesn't work the other way around

You should only use the regression line to make predictions for values of the dependent variable that are within the range of the given data > this is called interpolation (anything on the line)

Extrapolation = Making a prediction based on a value outside the range of the given data (anywhere else on the graph that isn't on the line)

Probability

Mutually exclusive events: P(A or B) = P(A) + P(B)

Independent events: P(A and B) = P(A) x P(B)

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Binomial Distribution

Can model X with binomial distribution X~B (n,p) if:

There are a fixed number of trials, n

2 possible outcomes (success and failure)

There is a fixed probability of success, p

The trials are independent of each other

n = sample size , p = probability

BPD for =

BCD for less than or equal to and great than or equal to

InvB > Area > X is less than or equal to just the number they give you , if X is more than or equal to use 1- the number given to you as the area

Hypothesis testing:

Result of the experiment or the statistics that is calculated from the sample called the test statistic

Null hypothesis H0, is the hypothesis that you assume to be correct H0: p =

Alternative hypothesis H1, tells you about the the parameter if your assumption is shown to be wrong H1: p =

One tailed, 2 tailed

Assume that the null hypothesis is true then if it is less than the given threshold called the significance level of the test then you reject the null hypothesis ( You are told what level to use)

Critical region is a region of the probability distribution which if the test statistic falls within it you would reject your null hypothesis

Critical Value is the first value to fall inside of the critical region

Acceptance region is a region of the probability distribution which if the test statistic falls with with it it would cause you to accept your null hypothesis

Actual significance level is the probability of the test statistics falling within critical region