Inferential Statistics

• Choosing a Statistical Test:

  Carrots Should Come, Mashed With Swede, Under Roast Potatoes.

  NOI, URR

  
• Type 1 Errors:

  • Optimistic error.
  • This is the incorrect rejection of a null hypothesis which is true.
  • A false positive.
• Type 2 Errors:

  • The failure to reject the null hypothesis that is false.
  • A false negative.
• Measures of Central Tendency: Mean:

  • The average of all data points.
  • Strengths:
  • Includes all data points, provides a precise average.
  • Weaknesses:
  • Can be distorted by extreme values (outliers).
  • Not ideal for skewed distributions.
• Measures of Central Tendency: Median:

  • Strengths:
  • Reflects the central point of the data.
  • Not affected by outliers, better for skewed data.
  • Weaknesses:
  • Does not account for all data points.
  • May not be as accurate in symmetrical data.
• Measures of Central Tendency: Mode:

  • Strengths:
  • Useful for nominal/categorical data.
  • Simple to calculate.
  • Weaknesses:
  • May not represent the data accurately if there is no clear mode or multiple modes.
• Measures of Dispersion: Range:

  • Strengths:
  • Easy to calculate, gives a quick sense of the spread of data.
  • Weaknesses:
  • Sensitive to outliers.
  • Doesn’t reflect the distribution of the majority of data points.
• Measures of Dispersion: Standard Deviation:

  • Strengths:
  • More precise than the range.
  • Gives a better measure of spread.
  • Considers all data points.
  • Weaknesses:
  • More complex to calculate.
  • Can be affected by outliers.
  • May not be meaningful in highly skewed data.
• Graphical Representation: Bar Charts:

  • Strengths:
  • Simple to interpret.
  • Useful for comparing categorical data.
  • Weaknesses:
  • Can become cluttered with too many categories.
  • May lose accuracy in large datasets.
• Graphical Representation: Histograms:

  • Strengths:
  • Ideal for continuous data.
  • Shows frequency distribution clearly.
  • Weaknesses:
  • Less precise with small datasets.
  • Not suitable for categorical data.
• Graphical Representation: Frequency Polygons:

  • Strengths:
  • Clearly shows trends and comparisons between distributions.
  • Weaknesses:
  • Less detailed than histograms.
  • Can be hard to interpret with complex data.
• Graphical Representation: Pie Charts:

  • Strengths:
  • Good for showing proportions.
  • Easy to understand for simple datasets.
  • Weaknesses:
  • Not suitable for large datasets.
  • Can be misleading if too many categories are used.
• Graphical Representations: Scattergrams:

  • Strengths:
  • Clearly shows correlations between two variables.
  • Visually simple.
  • Weaknesses:
  • May not reveal causation.
  • Can be hard to interpret without a clear relationship.
• Nominal Data:

  • Data that consists of categories with no particular order or ranking.
  • Gender, nationality, hair colour, eye colour.
  • Strengths:
  • Easy to collect and categorize.
  • Useful for classifying and counting frequencies.
  • Weakness:
  • Does not allow for any meaningful numerical operations (like averages).
  • Limited in terms of statistical analysis.
• Ordinal Data:

  • Data that consists of categories with a meaningful order, but the intervals between the categories are not equal or defined.
  • Likert scales, class rankings.
  • Strengths:
  • Allows for a ranking of items, giving more information than nominal data.
  • Easier to analyse than nominal data, as it shows relative positions.
  • Weaknesses:
  • The distances between categories are not uniform, making it harder to make precise comparisons.
  • Lacks the ability to perform complex mathematical calculations.
• Interval Data:

  • Data Interval data consists of ordered data with equal intervals between values, but there is no true zero point (the zero is arbitrary).
  • Temperature in Celsius or Fahrenheit, IQ scores, calendar dates.
  • Strengths:
  • Equal intervals allow for meaningful comparisons between values (e.g., the difference between 10°C and 20°C is the same as between 30°C and 40°C).
  • Allows for a wide range of statistical operations (e.g., mean, standard deviation, correlation).
  • More precise than ordinal and nominal data.
  • Weaknesses:
  • Lack of a true zero means you can't make statements about "absolute" values (e.g., 0°C doesn't mean "no temperature").
  • Mathematical operations involving ratios (e.g., "twice as much") are not meaningful because of the lack of an absolute zero.
  • Can be difficult to interpret in some contexts (e.g., comparing temperature scales like Celsius and Fahrenheit).
• What is Nominal Data?

  Data in categories.