Statistics in research

Introduction

• Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, and presentation of data.

• Data is sampled from a population and used to make inferences about the population.

• It is a fundamental tool in research.

Statistics in research

• Statistics is used to summarize data.

• It is used to make inferences about populations.

• It is used to make informed decisions.

• It is used to test hypotheses.

(Descriptive) Statistics

• Descriptive statistics is used to summarize data.

• It is used to describe the main features of a dataset.

• It is used to present data in a meaningful way.

• It is used to identify patterns in data.

(Descriptive) Statistics

Measures of central tendency

• Mean: Average value of a dataset.

• Median: Middle value of a dataset.

• Mode: Most frequent value in a dataset.

(Descriptive) Statistics

• The following is taken from Cooper (1968).

(Descriptive) Statistics

Measures of central tendency

(Descriptive) Statistics

Measures of central tendency

(Descriptive) Statistics

Measures of central tendency

(Descriptive) Statistics

Measures of dispersion

• Range: Difference between the maximum and minimum values.

• Variance: Average of the squared differences from the mean.

• Standard deviation: Square root of the variance.

• Interquartile range: Difference between the 75th and 25th percentiles.

(Descriptive) Statistics

Measures of dispersion

"Measures of dispersion for X"

1×3 Matrix{String}:
 "variance"  "std"  "range"

1×3 Matrix{Float64}:
 47.9362  6.9236  28.13

" "

"Measures of dispersion for Y"

1×3 Matrix{String}:
 "variance"  "std"  "range"

1×3 Matrix{Float64}:
 0.048988  0.221332  1.01

(Descriptive) Statistics

Data visualization

• Scatter plot: Relationship between two variables.

• Histogram: Distribution of a variable.

• Box plot: Distribution of a variable, quartiles.

• Density plot: Distribution of a variable, smoothed.

(Descriptive) Statistics

Box plot

(Descriptive) Statistics

Density plot

(Inferential) Statistics

• Inferential statistics is used to make inferences about populations.

• It is used to test hypotheses.

• It is used to make informed decisions.

• It is used to estimate parameters.

(Inferential) Statistics

Hypothesis testing

• Null and Alternative hypothesis.

• Types of error (Type I and Type II).

• P-value.

• Confidence interval.

(Inferential) Statistics

Null and Alternative hypothesis

• Null hypothesis: No effect or no difference.

• Alternative hypothesis: Effect or difference.

• Example: Null hypothesis: The vaccine has no effect. Alternative hypothesis: The vaccine has an effect.

(Inferential) Statistics

Types of error

• Type I error: Rejecting the null hypothesis when it is true.

• Type II error: Failing to reject the null hypothesis when it is false.

• Example: Type I error: Jail an innocent person. Type II error: Free a guilty person.

(Inferential) Statistics

P-value

• The probability of observing the data given that the null hypothesis is true.

• It is used to test hypotheses.

• (For historical reasons) It is compared to a threshold, usually 0.05.

(Inferential) Statistics

P-value

(Inferential) Statistics

Confidence interval

• A range of values that is likely to contain the true value of a parameter.

• It is used to estimate parameters.

• (For historical reasons) It is usually set at 95%.

(Inferential) Statistics

Confidence interval

(Inferential) Statistics

Confidence interval

(Inferential) Statistics

Confidence interval

Do and don’ts of statistics in research

• Do use the right measure of central tendency.

• Don’t use the mean when the data is skewed or has outliers.

• Do use the right measure of dispersion.

• Don’t use the variance when you have outliers.

• Do use standard deviation to preserve the units of the data.

Do and don’ts of statistics in research

• Don’t say we proved the hypothesis.

• Do say the data supports the hypothesis.

• Do report confidence intervals.

• Don’t confuse improbability with impossibility.

Biases in statistics

• Selection bias: When the sample is not representative of the population.

• Confirmation bias: When we look for evidence that confirms our beliefs.

• Publication bias: When only significant results are published.

• Extrapolation bias: When we extrapolate beyond the data.

• Causation bias: When we confuse correlation with causation.

Conclusion

• Ask questions, use PhD consult.

• Consider joining the Design and Analysis of Experiments PhD Course.