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Statistics for Research

In statistics, formulas are mathematical expressions used to calculate measures such as central tendency, dispersion, probability, and inferential statistics.

Why is it important?

Choosing the right fomula:

  • Ensures accuracy in results
  • Avoids producing incorrect significance levels or false relationships.
  • Supports research and decision-making rely on valid statistical methods.
  • Matches the type of data and study design.
  • Improves reliability and reproducibility

How to select the right formula

  1. Classify your data (nominal, ordinal, continuous).
  2. Define the research objective (description, comparison, correlation, prediction).
  3. Choose an appropriate statistical test based on your study design (paired or unpaired, one group or multiple groups).
  4. Check the assumptions (e.g., normality, variance equality).
  5. Select the formula that fits your analysis and ensure it matches the data type and assumptions.

Recommended resources:

TileStats. (2021). How to choose an appropriate statistical test.

  Cases Controls
Positive A B
Negative C D
Odds Ratio (OR)
  • Definition: Compares the odds of an event happening in one group to the odds in another group.
  • Why to use: Used in case-control studies to assess the strength of association between exposure and outcome.
  • Formula:
  • Online calculator
Relative Risk (RR)
  • Definition: The ratio of the risk of an event in the exposed group to the risk in the control group.
  • Why to use: Used in cohort, cross-sectional and RCT studies to determine how much more (or less) likely an event is in the exposed group.
  • Formula:  

  • Online calculator
Experimental Event Rate (EER)
  • Definition: The proportion of subjects in the experimental group that experience the event of interest.
  • Why to use: Useful for measuring the event rate in the treatment group.
  • Formula: 

  • Online calculator
Control Event Rate (CER)
  • Definition: The proportion of subjects in the control group that experience the event of interest.
  • Why to use: Measures the event rate in the control group for comparison with the experimental group.
  • Formula: 

  • Online calculator
Relative Risk Reduction (RRR)
  • Definition: The proportional reduction in risk between the control and treatment groups.
  • Why to use: Useful for understanding the relative decrease in risk with an intervention.
  • Formula: 

  • Online calculator
Absolute Risk Reduction (ARR)
  • Definition: The absolute difference in event rates between the control and experimental groups.
  • Why to use: Provides the actual decrease in risk with treatment.
  • Formula: 

  • Online Calculator
Absolute Risk Increase (ARI)
  • Definition: The increase in the event rate for an intervention compared to a control, used when the intervention causes harm.
  • Why to use: Used to measure the increase in adverse events with treatment.
  • Formula: 

Number Needed to Treat (NNT)
  • Definition: The number of patients that need to be treated to prevent one additional adverse event.
  • Why to use: Important in clinical decision-making to assess the effectiveness of treatment.
  • Formula: 

  • Online Calculator
Number Needed to Harm (NNH)
  • Definition: The number of patients exposed to a risk factor or treatment needed to cause one additional adverse event.
  • Why to use: Important in assessing the potential harm of a treatment.
  • Formula: 

Prevalence
  • Definition: The proportion of a population that has a condition at a specific point in time.
  • Why to use: Provides an estimate of how widespread a disease or condition is within a population.
  • Formula: 

  • Online Calculator
Incidence
  • Definition: The rate of new cases of a condition over a given time period.
  • Why to use: Measures how quickly new cases of a disease are developing in a population.
  • Formula: 

  • Online Calculator
Cumulative Incidence
  • Definition: The proportion of people who develop a condition over a specified period of time.
  • Why to use: Indicates the risk of developing a disease within a defined time frame.
  • Formula: 

  • Online Calculator
Case Fatality Rate (CFR)
  • Definition: The proportion of diagnosed cases of a disease that result in death.
  • Why to use: Helps understand the lethality of a disease within a given period.
  • Formula: 

Crude Mortality Rate
  • Definition: The total number of deaths from all causes in a population during a specified time period.
  • Why to use: Measures the overall risk of death in a population.
  • Formula:  
     
  • Online Calculator
Confidence Interval
  • Definition: A range of values that estimate an unknown population parameter, such as the mean, with a certain level of confidence. 
  • Why use it: Provide a measure of uncertainty around a sample statistic (e.g., a mean) and help infer the population parameter by accounting for sample variability. They are commonly used to assess the reliability of estimates from sample data.
  • Formula: 

     

    x̄  = sample mean
    Z = Z-score corresponding to the confidence level (e.g., 1.96 for 95% confidence)
    σ = population standard deviation
    n = sample size
  • Example: Suppose you measure the heights of 100 individuals and find a sample mean height of 170 cm, with a standard deviation of 10 cm. To find a 95% confidence interval for the true population mean: CI = 170 ± (10 /√ 100) = 170 ± 1.96 x 1 = 170 ± 1.96 
  • So, the 95% confidence interval is 168.04 cm ≤ μ ≤171.96 cm
  • Online Calculator
Correlation
  • Definition: A measure of the linear relationship between two variables.
  • Why use it: To assess the strength and direction of the relationship between variables.
  • Formula: (Pearson Correlation Coefficient)

    r  = correlation coefficient
    xi = values of the x-variable in a sample
    x̄ = mean of the values of the x-variable
    yi = values of the y-variable in a sample
    ȳ = mean of the values of the y-variable
  • Example: A correlation coefficient r=0.8 indicates a strong positive correlation.
  • Online Calculator
Hazard Ratio (HR)
  • Definition: A measure of how often a particular event happens in one group compared to another over time.
  • Why use it: Often used in survival analysis to compare the risk of events between groups.
  • Formula: 

    h1(t) = the hazard in the treatment group
    h0(t) = the hazard in the control group.
  • Example: An HR of 0.7 means the treatment group has 30% less risk of the event compared to the control.
  • Online Calculator
Mean
  • Definition: The average of a set of numbers.
  • Why use it: To summarise a dataset with a single value representing the centre.
  • Formula: 

    xi = each data point
    N = number of data points.
  • Example: The mean of [10, 12, 8, 14] is (10+12+8+14)/4=11
  • Online Calculator
Median
  • Definition: The middle value in a sorted dataset.
  • Why use it: To represent the central tendency, especially when data is skewed.
  • Formula: No specific formula; it's the middle value in ordered data.
  • Example: In the dataset [5, 7, 9, 10, 15], the median is 9.
  • Online Calculator
Normal Distribution
  • Definition: A probability distribution where data is symmetrically distributed around the mean.
  • Why use it: Many statistical methods assume normal distribution for testing hypotheses.
  • Formula (Probability Density Function):

  • μ = the mean
    σ = the standard deviation.
  • Example: A normal distribution with μ=0 and σ=1 is the standard normal distribution.
  • Online Calculator
Percentile
  • Definition: A measure indicating the value below which a given percentage of observations fall.
  • Why use it: To describe the relative standing of a value in a dataset.
  • Formula: No fixed formula, but the percentile rank is typically calculated from sorted data.
  • Example: If a test score is in the 90th percentile, 90% of people scored lower.
  • Online Calculator
p-Value
  • Definition: The probability of obtaining test results at least as extreme as the results observed, assuming that the null hypothesis is true.
  • Why use it: To determine statistical significance in hypothesis testing.
  • Formula: Calculated from statistical tests (e.g., t-test, chi-square).
  • Example: A p-value of 0.03 means there is a 3% probability the observed result could occur by random chance if the null hypothesis is true.
  • Online Calculator
Sample Size
  • Definition: The number of observations or data points in a study.
  • Why use it: To ensure the study has enough power to detect a statistically significant effect.
  • Formula: (Cochran's Formula)

     
    Z = Z-value
    p= estimated proportion of the population
    e= desired margin of error (precision)
  • Example: n0 = (1.96)^2 .0.5. (1 - 0.5) / (0.05)^2 = 384.16 
  • Online Calculator
Simple Linear Regression
  • Defintion: Used to model the linear relationship between two variables.
  • Why use it:  To predict the value of a dependent variable based on the value of an independent variable and to understand the strength and direction of their relationship.
  • Formula: 


    y = outcome/disease
    a = baseline outcome value when exposure is 0
    b = Beta-coefficent. The effect of the exposure on the outcome.

  • Example y = 41.16 + 4.97(6) = 70.98
  • Online calculator
Standard Deviation (SD)
  • Definition: A measure of the amount of variation or dispersion in a set of values.
  • Why use it: To understand how spread out data points are from the mean.
  • Formula: 

     
    xi = each data point
    x̄ = mean
    N = the number of data points.
  • Example: For a dataset [10, 12, 8, 14], the mean x̄= 11. The SD is approximately 2.45.
  • Online Calculator
Standard Error (SE)
  • Definition: The standard deviation of the sample mean distribution.
  • Why use it: To estimate the accuracy of a sample mean relative to the population mean.
  • Formula: 

    SD= Standard deviation
    n= sample size.
  • Example: If SD=10 and n=25, SE = 10/√25 =2
  • Online Calculator
Z-Score
  • Definition: The number of standard deviations a data point is from the mean.
  • Why use it: To compare data points from different distributions or assess how unusual a data point is.
  • Formula:

    x = the value
    μ = the mean
    σ = standard deviation.
  • Example: For x=85, μ=80, and σ=5, the z-score is (85−80)/5= 1
  • Online Calculator