Statistics

Descriptive statistics: to summarise and describe data

  • The mean blood pressure in a sample of middle-aged men is 119mmHg

Inferential statistics: use data from samples to draw conclusions and make inferences about populations

  • Would it be true if all middle-aged men were studied?


Descriptive Statistics:

  • Graphical methods

  • Summary measures of location and spread

Appropriate methods depend upon the type of data collected


Types of data:

  • Categorical

    • Nominal (sex, hair colour)

    • Ordinal (oral cleanliness, social class)

  • Numerical/Quantitative

    • Continuous (age, height, weight, bp)

    • Discrete (number of decayed teeth, children, etc)


Summarising: most commonly use proportions or percentages


Frequency table:

Categories | Frequency | Proportions


Pie Charts


Bar Charts

Frequency/Category


Summarising numerical data:

  • Mean - average value. Use when data is approximately symmetrical

  • Median - middle value. Use for skewed data

  • Mode - most common value. Use when data is dominated by a single value.


Histogram - symmetrical data, skewed

Frequency density/X


Bar charts = categorical data

Histogram = numerical data


Mean:

(sum of all the data) / n


Median:

Put in ascending order


Variability:

The mean and the median are measures of the location for the central tendency of the data.

A measure of variability/spread for approximately symmetrical data is variable.

Variance = standard deviation squared = the sum of (X-mean) squared / (n-1)

Standard deviation gives a measure of variability on the same scale as the mean

Standard deviation = square root of variance = the square root of: the sum of (X-mean) squared / (n-1)


Normal Distribution

A probability distribution describes the probabilities of observing each value of a variable.

The normal distribution is a distribution which describes variables which tend to cluster around the mean

  • Bell-shaped graph

It has 2 parameters = mean and variance


u = mean

6 = Standard deviation

N(u, 6 squared)


Variability for Skewed Data

Spread of the data can be measured using IQR


Outliers:

Outliers are values which are incompatible with the rest of the data. They can be considered influential on results.


Descriptive Statistics:

  • Summarise and describe data

  • Proportions, percentages for categorical data

  • Mean and standard deviation for symmetrical numerical data

  • Median and IQR for skewed numerical data


Population: the entire group of people that we want to make inferences about

Parameters: quantities which describe this population (e.g. mean, SD, etc)

Sample: a smaller group randomly selected from the population, which we use to estimate the population parameters


Sample means follow a normal distribution if the original observations have a normal distribution or if the sample is large.


Bias

The sample should be selected randomly

A representative sample is one where every member of the population has an equal chance of being selected.

Examples of bias - self-selection, only selecting from a subgroup and response bias.


Every time we take a different random sample, we would get a different estimate of the mean. The variability of these estimates gives us information on how accurate they are.

Estimate the SD of all the possible estimates of the mean.

This is called the standard error of the mean.

Standard error = standard deviation / (the square root of n)

  • As the variability fo the data increases, the standard error increases

  • As the sample size increases, the standard error decreases


95% confidence interval gives a 95% probability that the true population parameter leis in this interval

95% confidence interval = estimated mean +- 1.96 x standard error


General Confidence Intervals

Can contruct confidence intervals around different types of estimates

For 95% confidence interval - estimated mean +- 2.58 x standard error