Statistics

Statistics:

Chapter 1-8

Chapter1: Data Collection

Population & Samples

Population

The whole set of items that are of interest

Sample

A selection of items taken from a population

Cheaper than a census

Advantages

Less time-consuming than a census

Fewer people have to respond

Less data to process than a census

Disadvantages

data may not be accurate

Sample may not be large enough to give information about small sub-groups in a population

Census

Measures every member of a population

Advantages

Disadvantages

Time-consuming

Expensive

Cannot be used when the testing process destroys the item

Hard to process large quantity of data

Completely accurate

Sampling Frame

A list of items in a population

Sampling Unit

A single item in the frame

Sampling

Random Sampling

Non-biased

Every member has an equalchance of being selected

Representative f the population

Simple Random

Each member is allocated a number and random number is picked through a random number generator/ by a hat

Need a sampling frame

Stratified

The sample is made up of the members from the categories in the same proportion as they are in the population

For example: 10% of each category is taken for a 10% sample of the whole population

Systematic

Need a sampling frame

All the items in the sampling frame are ordered and then members of the list are selected at reular intervals. The first member is randomly selected

Advantages

Disadvantages

Easy to use on small populations

Might not be representative of the data

Need a sampling frame

Advantages

Disadvantages

If the sampling frame is not random then it can introduce bias

You need a sampling frame

Suitable for larger samples and populations

Quick

Simple

Advantages

Representative

Disadvantages

You would need a type of sampling to select members from each category.Which would therefore give the disadvantages of that sampling technique

Must have categories

Time-consuming

Non-Random Sampling

Quota

Oppurtunity

You choose members of the sample as you carry out the survey

Advantages

easy to carry out

Cheap

Disadvantages

May not be representative

highly dependent on the researcher

Te researcher selects a sample that reflects the characteristics of the population, You would choose each member through opportunity sampling

An interviewer would meet people and allocate them into the appropriate category . This continues until all the spaces in the category has been filled. If a person fits into a category which has been filled you simply just ignore them or if the person refuses to be interviewed you ignore them and find the next person

No sampling frame needed

Representatitve

Quick

Easy

Cheap

Allows for easy comparison between different groups within a population

Could be bias

Non-responses are not recorded as such

Population must be divided into groups which can be costly or inaccurte

Increasing scope of study increases number of roups which adds time and expense

Advantages

Disadvantages

Types of Data

Continuous Data

Discrete Data

Qualitative Data

Quantitative Data

Time - it can be 2.01s / 2.1s / 2 seconds

Data that can only be one particular value

Number of siblings

Non-numerical data

Eye colour

Numerical data

Shoe size

Data that can take any value within a particular range

Eexcel's Large Data Set

Chapter2: Measures of Location and Spread

Measures of Central Tendancy

Measure of loaction

A single value which describes a position in a data set

Measure of central tendancy

A single value which describes the centre of the data

Mode

The value that occurs most often

Median

The midle value when the data values are orered

Mean

symbol for mean: x with a line on top

Σx/n

Sum of all data values/ number of data vaues

Other Measures of Location

Lower Quartile

The data value one quarter of the way through the data set

Upper Quartile

The value three quarters of the way though the data set

Q1

Q3

Q2

Split the data set into 100 parts

Percentiles

For continuous data: n/4th value

For discrete data: n+1/4th value

Interpolation

When data is in a grouped frequency table and you assume that the data values are even distributed within each class to estimate the medians, quartiles or percentiles

Measures of Spread

Range

The difference between the largest and smallest value

Interquartile range

The difference between the upper and lower quartile

Interpercentile range

The difference between the values for two given percentiles

Variance and Standard Deviation

Variance

Measure how spread out a set of data is

Square root of the variance

Standard Deviation

σ²

Σfx²/Σf - mean²

σ

√(Σfx²/Σf -mean²)

Sxx/n

Sxx

A summary statistic used to make an equation easier to learn

Σx² - (Σx)²/n

√(Sxx/n)

Measure how spread out a set of data is

Coding

If the data is coded using the formula

y = x-a/b

the mean of the coded data

ȳ = x̄ - a/b

The standard deviation of the coded data

σy = σx/b

Chapter4: Correlation

Correlation

Bivariate data

data which has pairs of values for two variables

Describes the linear relationship between two variables

Causal relationship

if a change in one variable causes another hange in another variable

Negative

Strong

Weak

Positive

Strong

Weak

Linear Regression

Least Squares Regression Line

A type of line of best fit which minimises the sum of the squares of the distances of each data point from the line

The regression line of y on x

written as y = a + bx

The coefficient b tells you the change in y for each unit change in x

The regression line for y on x will be different to the regressionline for x on y

usually just called the regression line

You should only use the reression line to make predictions for values of the dependent varable that are within the rane of data given. If you need to predict the independent variable/y then you need to use the regression lne x on y

Chapter3: Representations of Data

Outliers

An extreme value that lies outside the pattern of dat

greater than Q3 + k( Q3-Q1)

Less than Q1 - k( Q3- Q1)

anomaly

An outlier that is a mistake

When you take out an anomalous result it is know as cleaning the data

Historams

Grouped continuous data can be represented in a histogram

Area of the bar is proportionate to the frequency

frequency density= frequency/class width

If you join the middle of the tops of each bar in a histogram with equal class widths forms a frequency polygon

Comparing Data

When commenting on data sets you can comment on a measure of location or spread

You can compare data sets on median, mean, IQR or standar deviation

If the data sets contain extreme values then you compare data sets n the median or IQR

Box Plots

An outlier is shown by a cross

Cumulative Frequency