Distributions (College Board AP® Psychology): Revision Note

What is a distribution?

• Distribution refers to the way in which scores are spread across a data set

  • How scores cluster around the mean and whether that spread is symmetrical or asymmetrical

• Researchers in psychology are interested in the shape of a distribution because it affects:

  • which measure of central tendency is most appropriate

  • how standard deviation should be interpreted

  • what conclusions can be drawn from the data

• There are three key distribution shapes:

  • Normal distribution — symmetrical, bell-shaped

  • Skewed distribution — asymmetrical, with one tail longer than the other

  • Bimodal distribution — two distinct peaks

Normal distribution

• A normal distribution is a symmetrical, bell-shaped distribution in which most scores cluster around the mean, with progressively fewer scores toward the extremes

• In a perfect normal distribution:

  • the mean, median, and mode are all equal and located at the peak of the curve

  • scores are distributed symmetrically on both sides of the mean

  • the curve is highest in the middle and tapers gradually toward both ends

• The shape of a normal distribution is known as the bell curve

• Examples of data that tends to be normally distributed:

  • Height

  • Weight

  • IQ scores

  • Reaction time

Percentage benchmarks in a normal distribution

• One of the most important features of the normal distribution is that specific percentages of scores fall within each standard deviation of the mean:

  • Approximately 68% of scores fall within 1 standard deviation of the mean

  • Approximately 95% of scores fall within 2 standard deviations of the mean

  • Approximately 99% of scores fall within 3 standard deviations of the mean

• This means that scores falling beyond 2 standard deviations from the mean are relatively rare

  • The occur in only about 5% of cases

• Example:

  • IQ scores have a mean of 100 and a standard deviation of 15

  • Approximately 68% of people have an IQ between 85 and 115

    • This is within 1 SD of the mean

  • Approximately 95% of people have an IQ between 70 and 130

    • This is within 2 SDs of the mean

  • A score beyond 2 standard deviations from the mean (below 70 or above 130) would be considered unusually low or high

Link to z-scores

• Because the normal distribution has predictable percentage benchmarks, z-scores can be used to determine exactly where a given score falls within the distribution

  • A z-score of +1 corresponds to approximately the 84th percentile

    • The score is higher than approximately 84% of all scores

  • A z-score of −1 corresponds to approximately the 16th percentile

    • The score is higher than only approximately 16% of all scores

  • A z-score of +2 corresponds to approximately the 98th percentile

The normal distribution and deviance

• The normal distribution can be used to identify scores that deviate significantly from the norm

• Scores that fall beyond 2 standard deviations from the mean may indicate an unusually high or low result, e.g.:

  • A very high score on an IQ test

  • A very high score on a depression scale following childbirth

  • A very low score on an empathy scale

Skewed distributions

• A skewed distribution is one in which scores are not distributed symmetrically around the mean

  • One tail of the distribution is longer than the other

• Skewed distributions occur when there are behaviors, conditions, or test scores that do not fit neatly into a normal distribution

• In a skewed distribution:

  • The mean, median, and mode no longer have the same value and are no longer located at the same point on the curve

  • The mean is the measure of central tendency most affected by skew

    • Because it takes all scores into account, it is pulled toward the extreme scores in the longer tail

  • The median is less affected by skew and is therefore a more appropriate measure of central tendency for skewed data sets

  • The mode remains at the peak of the distribution

Positive skew

• A positively skewed distribution is one in which most scores cluster toward the left (lower end) of the distribution, with a long tail extending to the right (higher end)

  • The mean is pulled to the right of the median and mode by the extreme high scores in the tail

• In a positively skewed distribution: mode < median < mean

• Examples of positively skewed data:

  • Scores on a very difficult exam

    • Most students score at the lower end, with only a few achieving high scores

  • Income distribution

    • Most people earn relatively modest incomes, with a small number of very high earners pulling the tail to the right

  • Reaction time data

    • Most responses are fast, but occasional very slow responses create a long right tail

Negative skew

• A negatively skewed distribution is one in which most scores cluster toward the right (higher end) of the distribution, with a long tail extending to the left (lower end)

  • The mean is pulled to the left of the median and mode by the extreme low scores in the tail

• In a negatively skewed distribution: mean < median < mode

• Examples of negatively skewed data:

  • Scores on a very easy exam

    • Most students score at the higher end, with only a few achieving very low scores

  • Age at retirement

    • Most people retire at a similar older age, with a small number retiring unusually early creating a left tail

  • Hours of sleep in a healthy adult population

    • Most adults sleep close to 8 hours, with a small number sleeping very few hours pulling the tail to the left

Bimodal distribution

• A bimodal distribution is one that has two distinct peaks

  • Two values that occur with equal or near-equal frequency, both more often than any other value in the data set

• A bimodal distribution suggests that the data set contains two distinct subgroups that responded differently, e.g.

  • A memory test administered to both young adults and older adults may produce a bimodal distribution

    • One peak representing the younger group's scores and one peak representing the older group's scores

  • A survey on attitudes toward a controversial policy may produce a bimodal distribution

    • One peak representing strongly favorable responses and one peak representing strongly opposed responses

• When a bimodal distribution is present:

  • the mean may fall between the two peaks

    • It would not represent either subgroup accurately

  • the mode alone is not sufficient as a summary statistic

    • Both peaks should be reported

  • the bimodal pattern itself is a meaningful finding that warrants further investigation into why two distinct clusters of scores exist