Statistics 2

Obesity Levels in England

• Obesity levels have risen among Year 6 children in England.
• Data shows a gradual upward trend in obesity rates among Year 6 children in England between 2006/07 and 2021/22.

Descriptive Statistics

• In the previous lecture, the foundations of descriptive statistics were covered.

Statistical Noise

• Real-world measurements are affected by noise or error.
• A graph of mean UK temperature from Jan 1 to Dec 31, 2023 displays daily fluctuations.

Drug Efficacy

• A bar graph displays the effectiveness of a treatment.
• The untreated group has a recovery rate of about 30%.
• The treated group has an efficacy rate around 70%.

Normal Distribution

• Statistical noise often follows a normal distribution.
• A normal distribution graph shows a bell curve shape.
• Data points are clustered around the mean, with fewer data points the further from the mean value.

Measures of Centrality & Dispersion

• The normal distribution shows a measure of centrality (i.e., the mean) and dispersion around it (i.e., the standard deviation)
• A mean (average), median (middle), and standard deviation (shows how much dispersed the data is) were discussed.

Z parameter

• The Z parameter is used to determine probabilities in a normal distribution.
• The Empirical Rule in a normal distribution was given in a chart showing percentages of data within +/- 1, 2, and 3 standard deviations.

Non-normal Distributions

• Non-normal distributions exist in real-world scenarios.
• A histogram displays the distribution of equivalised household disposable income.

Median and Interquartile Range

• The median is the measure of centrality.
• The interquartile range is the measure of dispersion for non-normal data.

Population vs Sample

• Descriptive statistics include definitions of population and sample.
• A population is all the people (or objects) you are interested in.
• A sample is a smaller collection of a population that measures population properties.

Sampling

• Populations are often too big or complex to measure.
• To avoid measuring the whole population, sampling is used.
• Samples are chosen to represent a whole population.

Standard Error of the Mean

• A standard error of the mean (SEM) is a reduced standard deviation.
• SEM is calculated if you want to measure standard deviations of multiple samples of the same population.

Inferential Statistics

• Inferential statistics uses data from samples to make conclusions about larger populations.
• In inferential statistics, decisions about signal vs noise are determined.

Variables

• All statistical analyses involve predictor and outcome variables.
• A predictor variable is believed to cause an effect.
• An outcome variable is believed to show an effect.

Variables: Categorical & Continuous

• Variables can be categorical or continuous.
• Categorical variables use labels without order (e.g., sex, country).
• Continuous variables use ordered numbers (e.g., blood pressure, age).

Hypotheses

• Hypotheses are important aspects of all inferential statistics.
• The null hypothesis (H₀) posits no significant effect (just noise).
• The alternative hypothesis (H₁) indicates a significant effect (i.e., a signal).

P-value

• The p-value is a probability used in statistical tests.
• It shows the probability the effect is merely random chance.
• A low p-value suggests a statistically significant effect.

P-value Significance

• A p-value of less than 0.05 (5%) usually means a significant result.

Inferential Statistics: Cases

• Statistical tests are categorized into cases based on variables.

Correlation

• This is the case when predictor and outcome variables are both continuous.
• Correlation shows a relationship between variables.
• Correlation can be positive, negative, or nonexistent.

Pearson's Correlation Coefficient

• The Pearson's correlation coefficient (r) quantifies the strength and direction of a linear correlation.
• The value of r is from -1 to +1.
• Values closer to -1 or +1 are for strong correlations.

Correlation is not Causation

• Correlation tells if variables are linked but not if one causes the other.
• A third variable may affect both variables.

Spearman Rank Correlation Coefficient

• The Spearman Rank Correlation Coefficient is used when the relationship isn't linear.

Correlation Testing

• This session's activities include testing an example using data to discover if there was a warming trend (i.e., a correlation between time and temperature).