Descriptive Statistics

  • Descriptive statistics are used to summarize data
  • To obtain and define a measure, some type of numerical value must be associated with the measure
  • One method of classifying numbers in measurement is by using numerical scales

Scales of Measure Hierarchy

  • Nominal - classification scale with NO order, magnitude, or size
  • Ordinal - a scale that provides information of the rank order of the scores, but no magnitude or differences can be determined
  • Interval - a scale that provides both order and magnitude, but has no true zero point
  • Ratio - a scale that possesses all of the characteristics of the interval scale and has a true zero point

Measures of Central Tendency

  • Measures of central tendency describe the distribution of scores around a midpoint
  • Most statistical methods for data analysis assume that the distribution of scores is normal
  • A normal distribution is graphically represented by a normal (bell shaped) curve
  • The three descriptive statistics used to determine central tendency are the mean, median and mode

The Mean

  • The Mean - the most commonly used measure of central tendency, refers to the arithmetic average of a group of scores
  • It is the most sensitive measure of central tendency
  • It is the most appropriate measure of central tendency when using ratio data
  • It considers all information about the the data
  • It is influenced by extreme scores, especially in small sample

The Median

  • The median is the score that represents the exact middle in a distribution
  • It is not affected by extreme scores, so it is a more representative of the central tendency when extreme scores are present
  • It is a measure of position within the distribution
  • It is NOT used for additional statistical calculations

The Mode

  • The mode is the score within a distribution that occurs most frequently
  • It is the least used measure of central tendency
  • It is not used for additional statistical calculations
  • It is the most unstable measure of central tendency

Distribution Shapes

  • The graphical shape of a distribution can tell a person about the distribution of scores and about the sample being measured
  • The distribution shape can be defined by symmetry or by height and width
  • The term for the symmetry of a distribution is skewedness, while the height and width is described by kurtosis

Skewed Distributions

  • A distribution can be positively skewed, normal, or negatively skewed.
  • Skewedness is described by the where scores cluster within a distribution

Kurtosis

  • Distributions can also be described by height and width
  • Leptokortic means that the data are very similar (homogeneous),
  • Platykurtic means the data are very different (heterogeneous)
  • Bimodal data cluster at both ends of the distribution

Variability

  • A curves kurtosis is the general introduction into the discussion of variability
  • The degree or spread of scores is defined as the variability
  • Three types of variability
  • Range
  • Semi-Interquartile range
  • Variance

Variance

  • The variance is the squared deviation of each score from the mean, or for lack of a better example, the standard deviation squared (s2)
  • The measured or observed variance is the sum of the true variance and the error variance.

Standard Scores

  • To put variance into perspective, we can determine standard scores within a distribution
  • Standard scores are scores that are standardized around a mean.
  • The two standard scores we will discuss are the z-score and T-score

The z-Score

  • The z-score for an individual gives us an estimate of how many standard deviations an individual’s score fall from the mean
  • The z-score is simply the difference between an individual’s score and the mean divided by the standard deviation

The T-score

  • The T-score standardizes scores around a mean of 50
  • The T-score is simply = 50 + 10*(z-score).
  • Either standardized score (z or T) is a way of determining where an individual’s score falls from the mean.
  • In a normal distribution, you can determine the percentile for each scores

Statistics

  • The correlation coefficient - determines the relation between scores and variables
  • Linear regression - predicts future outcomes based on the correlation between independent and dependent variables.

Inferential Statistics

  • A method of hypothesis testing
  • The null hypothesis and research hypothesis
  • Types of error; Type 1 (alpha) and type II (beta)
  • Types of variable; independent vs dependent

Possible Test Items From These Notes:

  • Be able to define and differentiate between the types of statistics, scales, measures of central tendency, and distributions.
  • Be able to define variance, the types of variance, and how variance is used in a normal distribution.
  • Know the different standardized scores available, how they are connected to a normal distribution, and what interpretations can be made with standardized scores.
  • Be able to define and differentiate between standard deviation, variance, and correlation.
  • Know the difference between independent and dependent variables, types of errors, and types of hypotheses.
  • Be able to describe probability and its importance in evaluation, the usage of the different statistical tests and with which types of scales of measure are used with each.