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Transcript
Statistics
Josée L. Jarry, Ph.D., C.Psych.
Introduction to Psychology
Department of Psychology
University of Toronto
June 9, 2003
Statistical Procedures
• Mathematical aids for summarizing and
interpreting data
• Descriptive statistics
– Used to summarize data sets
• Inferential statistics
– Used to determine what conclusion can be
drawn from data sets
Organizing and Summarizing
Scores
Frequency distribution
Central tendency
Variability
Frequency Distribution
• Rank order
– Organize scores from lowest to highest
• Frequency distribution
– Divide the range of scores into equal intervals
– Determine how many scores fall into each
interval
• Demonstrates the frequency of occurrence
of each response or range of responses
Shapes of Frequency Distributions
Normal Distribution
• Maximum frequency lies in the center of
the range of scores
• Frequency tapers off symmetrically on
both sides
• Many measures in nature are normally
distributed
• Found when a measure is determined by
several independent factors
Bimodal Distributions
• Occurs when scores form two separate
groupings
• Mode
– the most frequently occurring score or range of
scores in a frequency distribution
• Two separate areas of peak frequencies or
two modes
• The normal curve is unimodal
Skewed Distributions
• Positively skewed distribution
– Spread of scores above the mode is greater than
the spread below
– The tail extends in the direction of high scores
• Negatively skewed distributions
– Spread of scores below the mode is greater than
the spread above
– The tail extends in the direction of low scores
Measures of Central Tendency
• Consists of summarizing an entire
distribution with a single score that
represents the centre of the distribution.
• Median
– Is the middle score of a set of ranked scores.
• Mean
– Found by adding all the scores and dividing by
the total number of scores.
• M = sum of score
N
Mean, Median, & Distributions
• In a normal distribution
– Mean and median are identical
• In a positively skewed distribution
– The mean is greater than the median;
• In a negatively skewed distribution
– The mean is smaller than the median
• Mean is preferred in normal distributions
• Median is preferred in highly skewed
distributions
Measures of Variability (1)
• Measures of variability tell us how widely
the observations are spread around the
centre
• Two distributions can both be normal and
have the same mean, but have very different
variability.
Measures of Variability (2)
• Range
– The difference between the highest and the
lowest scores in a distribution
• Variance
– Takes into account the extent to which all of the
scores in the distribution differ from each other
• Standard deviation
– Expresses variability in the same unit of
measurement as the original scores
– The square root of the variance
Converting Scores for Purposes of
Comparison
• Allows to compare different kinds of scores
to each other
• To compare different kind of scores with
each other, each score must be converted
into a form that expresses directly its
relationship to the whole distribution of
scores from which it came
Percentile Rank
• The most straightforward way of comparing
one person to another on a given measure
• Allows comparison between different
measures
• Determine that person’s percentile rank on
the measures of interest
• Percentile rank
– the percentage of scores that are equal to that
score or lower, out of a whole set of scores for
that measure
Standardized Scores
• z-score
– Expresses a score in terms of the number of
standard deviations that the original score is
away from the mean of the original scores
• Convert any score into a z-score
– calculate its deviation from the mean
– divide the deviation by the standard deviation
of the distribution
–z=
score - mean
Standard deviation
Correlation Coefficient (1)
• Is a mathematical means of describing the strength
and direction of the relationship between two
variables that have been mathematically measured
• Varies between -1 and +1
• The sign (+ or -) of the correlation indicates the
direction (positive or negative) of the relationship
• The absolute value of the coefficient (from 0 to 1)
indicates the strength of the relationship
Correlation Coefficient (2)
• A positive correlation indicates that an
increase in one variable corresponds to an
increase in the other
• A negative correlation indicates that an
increase in one variable corresponds to a
decrease in the other
• Scatter plot
– Visual display of a correlation
– Each point represents a subject’s pair of scores