STATS OCR A LEVEL MATHS

Statistical Sampling:

What is a census and what are the downfalls

A question asked of the whole population

It is expensive, time consuming, and sometimes biased if certain people do not respond

Statistical Sampling:

Discrete data means

Distinct categories

Aka shoe sizes

Statistical sampling:

Continuous data means

Ever increasing levels of accuracy

Aka the exact length of your foot

Statistical sampling

Quantitative data

Has a numerical value

Statistical sampling:

Qualitative data

Is worded

Statistical sampling:

The Simple Random method

Downfalls + positives

-assign numbers to everything in a population

-select random numbers with a raffle or random number generator

-remove any repeats

Ideal for self-contained areas, but time-consuming with large populations

Statistical sampling:

Systematic method

Downfalls and positives

-Assign numbers to everything

-pick every nth person

Has a periodic nature so is useful for finding patterns

Could be biased if every nth item is the same.

Statistical sampling:

Stratified method

Downfalls and positives

Select a sample of n that proportionally represents all categories

Can be costly trying to find populations from different categories

Can focus attention on particular areas to see if that influences results

Statistical sampling:

Quota method

Downfalls and positives

Find people from two categories

Make them do the questionnaires to meet the quota

Not randomly selected, must fill quota before finishing sample

It is cost effective

Statistical sampling:

Opportunity method

Downfalls and positives

Stand in the street with a clipboard

Very biased

Self contained, fast, cost efficient, no quota

Statistical sampling:

Cluster method

Downfalls and positives

Split the samples of people into clusters (group similar items together)

Easy but there could be a key difference in the clusters you are comparing

Statistical sampling:

Self-selection

Volunteered responses

Cost efficient, fast

Can attract a certain type of opinionated people

Probability:

P(AuB)=

(P(A)+P(B))-P(AnB)

Probability:

When drawing a venn diagram, writing in values, start with

The central overlap

Probability:

P(AnB)=

(P(A)+P(B))-P(AuB)

Or use a venn diagram visually

Or P(A) x P(B)

Probability:

If two events are independent

P(AnB)= P(A) x P(B)

P(A|B)=P(A)

Probability:

Two events are mutually exclusive

P(AnB)=0

Probability:

P(A|B)=P(AnB)/P(B)

Probability:

For binomial distribution to be valid

There must be a fixed no. of trials

Two outcomes

Each event must be independent

Probability must be constant

Presentation:

Box plot

Has min, LQ, median, UQ, max

Graphs:

how to find outliers on a box plot

Upper outliers: UQ+1.5(IQR)

Lower outliers: LQ-1.5(IQR)

Graphs:

Comparing box plots

Compare medians to discuss average

Compare IQR to discuss consistency

Compare difference in min and max to discuss performance

Graphs:

Drawing cumulative frequency curves

Find the cumulative frequency by adding the frequencies going along the table of x values

Plot CF-x and find the curve

Graphs:

Finding the median, LQ and UQ of the cumulative frequency graph

Median: CF/2 so find the x-value at CF/2

LQ: CF/4 so find the x value

UQ: CF x ¾ so find the x value

Graphs:

Drawing a histogram

Find the frequency density for the table

Frequency density= f/class width

Plot fd-x

Area of the boxes= k (frequency)

Often k is 1 but check

Graphs:

Find median, LQ and UQ for a histogram

Find the the sum of the frequencies

find the sum of F/ 2 which equals a constant a

So the a^th value along is where the median lies

So do (X-lower part of inequality)/( upper - lower part of inequality)= (a-lower value of inequality)/(upper- lower value of inequality)

LQ: sum of F/4

UQ: sum of F x ¾

Bivariate data:

Definition and how to plot

Two variables, generally plotted with a scatter graph, with the independent variable horizontal and the dependent variable vertical

Bivariate data:

Correlation

-1<=r<=1

Where -1 is strong negative

1 is strong positive

0 is no correlation

Bivariate data

How to find PMCC

Use given formula in booklet

Sxy/ (sqrt Sxx Syy)

Or

Calculator:

Stats, x in left column, y in right column, CALC, REG, X, a+bx, r=…

Or

Guess between -1<=r<=1 looking at correlation

Bivariate data:

Regression line is .. and how to find it

An optimal line of best fit

y-y_=b(x-x_)

find x_ and y_ therefore the regression line must go through (x_,y_)

Find b on calculator using STATS, x first column, y second column, CALC, REG, X, a+bx, b=…

Draw the regression line using y=b(x-x_)+y_

Bivariate data:

Interpolation and extrapolation meaning

Interpolation: anything inside the region is valid

Extrapolation: extending the region to guess a value

Discrete data:

Find the mean, mode, median, variance from discrete data

Mean= sum of x . Frequency/ n

Calc: STATS, first column x, second column F, CALC, SET, 1 var freq: List 2, exit, 1-VAR, x_=…

Discrete data:

Find the mode from discrete data

Look at the highest frequency

Discrete data:

Find the median, LQ, UQ

Find the sum of the frequencies

(Sum+1)/2= position of the median

Find the CF column

Identify where the median is and find its value

LQ: (sum/2 +1)/2= position of LQ

UQ: sum/2 + position of LQ

Grouped data:

Estimate the mean

Estimate the mean is in the middle of the group

So find the midpoint for each inequality, use STATS, to find mean with midpoints in column 1 and frequency in column 2

If the inequalities aren’t continuous, rewrite so they are

With age always change the inequality aka 10<=x<=15 should be 10<=x<16

Grouped data:

Estimate the median, LQ, UQ

Find the cumulative frequency

Divide the cumulative frequency by 2 to find the position of the median

Find the median using linear interpolation

LQ: cumulative frequency/4

UQ: cumulative frequency x ¾

Comparing data sets:

What to say about median, IQR, variance

Median: discuss their average

IQR: discuss consistency

Standard deviation: discuss consistency

Variance and standard deviation:

Find the variance and s.d from a table of x values

Find the mean

Find the distance of each value from the mean

Square the distances

Find the sum of the squares of the distances

Divide by n=variance

So sqrt the variance= standard deviation

Use calc or given formula or remember this method

Standard deviation and variance:

Difference between ōx and sx

ōx is for population

sx is for a sample

linear coding:

Convert the mean and standard deviation from celsius to F’

Find the mean and standard deviation in terms of celsius

The mean should be effected by the multiplication and addition in the conversion formula

The standard deviation should only be multiplied or divided by the value in the conversion formula

Outliers:

Find outliers using quartiles

Find outliers using mean and standard deviation

Quartiles:

LQ-1.5(IQR)=outlier

UQ+1.5(IQR)=outlier

Mean and s.d:

mean-2(s.d)= outlier

mean+2(s.d)=outlier

Omit them if they are an outlier

Discrete random variable:

Tabulate data

X (possible outcomes)

P(X=x) (probabilities) sum should= 1

Discrete random variables:

Stick graph

Tabulate data

Y-axis is P(X=x)

x-axis is X (outcomes)

Discrete random variables:

Find data from tabulated data

Discrete random variables:

Notation P(X=x)=x/15, x=1,2,3,4,5

Probability X=x is x/15 when x=1,2,3,4,5

Cumulative distribution function:

Tabulate

x (outcomes)

P(X=x) (probabilities)

F(x) (cumulative probabilities) with the last value in the row being 1

x must be whole for discrete data!!