RMP Data reduction, samples and population

Categorical variables

where the different possible values that the variable can take are “unordered” (not numbers)

Measure of central tendency for categorical variables

Mode

Ordinal variables

where the different possible values that the variable can take are ordered (whole numbers)

the difference between 1^st and 2^nd place may not be the same as that between 3^rd and 4^th, etc

Measure of central tendency for ordinal variables

Median

Continuous variables

numeric data (can have decimals)

difference between 1^st and 2^nd place is the same as that between 3^rd and 4^th, etc

Ratio data

If the number 0 indicates the absence of the quantity under consideration

e.g., Money, Time to travel a distance

Interval data

If the number 0 does not indicate absence

e.g. temperature

Usual measure of central tendency for continuous variables

mean

If the distribution is asymmetric (skewed data), If histograms reveal outliers

When would you use the median as a measure of central tendency for continuous variables?

Mode

Most common element

Median

The score that lies in the middle of the sample, which therefore has an equal number of scores higher and lower than it

Mean

The average

Disadvantages of the mode

Can have more than one mode

May not be representative of your data

Advantages of the median

Unaffected by outlier scores

Unaffected by skewed distributions

Disadvantages of the median

Can fluctuate across samples

Less helpful with parametric statistics

Advantages of the mean

Represents the whole sample

Resistant to differences across samples

Disadvantages of the mean

Very influenced by extreme/ outlier scores

Dispersal

spread/ variability of the data

Measures of dispersal

Range

Boxplots

Inter quartile range

Inter quartile range (IQR)

difference between the 25^th and 75^th percentiles of the data

Outliers

1.5 x IQR

Reducing dispersion of data into single values

Variance

Standard deviation

Standard error

Variance

subtract each score from the mean

square them

add them up

divide by the number of scores minus 1

Standard deviation

Typical range of dispersion around the data

Calculating standard deviation

Calculate variance

Square root variance

Standard error

Find standard deviation

divide by the square root of the sample size

When to use standard error

samples taken from wider populations

What does standard error tell us

smaller samples are penalised, i.e. larger samples will return smaller values

give you a sense of how confident you are of what the mean is

Population

What you want to generalise to from your sample

Sampling distribution

given the sample mean/ standard deviation, what range of values might the unmeasured population fall into

Calculating sampling distribution

calculating the mean of each sample and examining the distribution

Central limit theorem

if you have a sample size of 30 or more, then the sampling distribution is always ‘normally distributed’, even if the population distribution is not

Normal distribution

bell shaped curve

symmetric figure

mean and median at around the same point

Confidence intervals

how confident you can be that if you collected a sample again it would fall within the intervals

Commonly report 95% confidence = 95% sure the mean would fall within these values if we collected a sample again

1 standard deviation

Either side of the mean captures 68% of the population

1.96 standard deviation

captures around 95% of the population '

commonly the point we state the data is unlikely to have come from the sample

Z scores

tells you how many standard deviations a raw set of data is above or below the normal distribution

Random sampling

every member of the population should have an equal chance of becoming part of the sample

Opportunity samples

e.g. an advert is put up, the type of people that respond to adverts may not be representative of the whole population 

Why larger samples are better

give us more confidence in generalising our sample mean to the population mean

more likely to lie close to the population mean

Standard normal distribution (Z)

Converts any normal distribution into a scale with a mean of 0 and a standard deviation of 1

Parametric analysis

assumes data fits a specific probability distribution (a normal distribution)

Why use Z scores

allow you to compare data points from different datasets by standardising the values

useful when the datasets have different scales or units

Calculating Z scores (population)

X (raw score) - μ (population mean)

divided by σ (population standard deviation)

Calculating Z scores (sample)

X (raw score) - x̄ (sample mean)

divided by S (sample standard deviation)