Measures of Dispersion (College Board AP® Psychology): Revision Note

What are measures of dispersion?

• Measures of dispersion calculate the spread of scores in a data set

  • This refers to how much scores vary from one another and how far they are from the central measure

• A data set with low dispersion has scores that cluster tightly around the measure of central tendency

  • There is little variation among scores

• A data set with high dispersion has scores that are spread widely apart from the central measure

  • There is considerable variation among scores

• If every participant in a data set scored identically (e.g. everyone scored 15 out of 20 on a memory test), the dispersion score would be zero

  • This is because there is no variation at all

• There are two measures of dispersion:

  • The range

  • The standard deviation

Range

• The range describes the difference between the lowest and highest scores in a data set

• The range provides a broad overview of the total spread of scores

How to calculate the range

• Subtract the lowest value from the highest value in the data set

• Example:

  • Data set: 4, 4, 6, 7, 9, 9

  • 9 − 4 = 5

  • Range = 5

How to interpret the range

• The range tells us the total spread of scores from the lowest to the highest value

  • A large range indicates that scores are widely spread

    • There is considerable variation between participants

  • A small range indicates that scores are clustered closely together

    • There is little variation between participants

• Example:

  • If the range of anxiety scores in an experimental group is 4 and the range in a control group is 22, this suggests that participants in the experimental group responded much more consistently than those in the control group

Evaluation of the range

Strengths

• The range is quick and easy to calculate

  • It provides an immediate overview of the total spread of scores in a data set

• The range is useful for identifying whether extreme scores are present in a data set

  • a very large range suggests that outliers may be present, which would then indicate that the median is a more appropriate measure of central tendency than the mean

Limitations

• The range only takes two scores into account

  • This means that it provides no information about how all the other scores in the data set are distributed

• The range is not very stable or representative

  • a single outlier can dramatically increase the range, giving a misleading picture of the overall spread of scores

Standard deviation

• Standard deviation measures how much the scores in a data set deviate from the mean

  • It provides insight into how clustered or spread out scores are around the mean

• Standard deviation is a more sensitive and informative measure of dispersion than the range because it takes every score in the data set into account

How to interpret standard deviation

• A low standard deviation indicates that scores are clustered tightly around the mean

  • This means that participants responded consistently and the data set is reliable

• A high standard deviation indicates that scores are spread widely around the mean

  • This means that there is considerable variation between participants and the data set is less consistent

• Example:

  • If the mean score on a depression scale is 18 with a standard deviation of 2, this tells us that most participants scored close to 18

    • The scores are tightly clustered and the data is consistent

  • If the standard deviation were 12, scores would be widely spread around the mean of 18, suggesting considerable variation in participants' responses

Link to normal distribution

• In a normal distribution, scores cluster symmetrically around the mean and standard deviation is relatively low

  • Most scores fall close to the mean with fewer scores at the extremes

• A high standard deviation suggests that the data may be more spread out or skewed, which affects how the findings should be interpreted

Evaluation of standard deviation

Strengths

• Standard deviation is more sensitive than the range as it uses all the scores in the data set 

  • This means that it is a more valid representation of the overall spread of scores

• Standard deviation provides meaningful information about the consistency and reliability of a data set

  • A low standard deviation indicates that findings are likely to be replicable

Limitations

• Standard deviation is more complex to calculate than the range

  • However, statistical tools make this straightforward in practice

• Standard deviation can be distorted by extreme outliers

  • A single extreme score may inflate or deflate the standard deviation

  • This gives a misleading representation of how spread out the scores are

Percentile rank & regression toward the mean

Percentile rank

• A percentile rank indicates the percentage of scores in a data set that fall at or below a given score

  • Percentile rank is used to show where an individual score sits relative to all other scores in the data set

• Example:

  • If a student scores at the 70th percentile on a standardized psychology exam, this means their score is equal to or higher than 70% of all students who took the exam

  • If a student scores at the 30th percentile, their score is equal to or higher than only 30% of all students

How to interpret percentile rank

• A high percentile rank (e.g. 90th percentile) indicates that the score is well above the majority of scores in the data set

• A low percentile rank (e.g. 10th percentile) indicates that the score falls below the majority of scores in the data set

• The 50th percentile corresponds to the median

  • Exactly half of all scores fall below this point and half fall above it

• Percentile rank and standard deviation are closely related

• In a normal distribution, knowing a participant's percentile rank tells you how far their score sits from the mean in terms of the overall spread of scores, e.g.:

  • A score at the 84th percentile sits approximately one standard deviation above the mean

  • A score at the 16th percentile sits approximately one standard deviation below the mean

• Percentile ranks are commonly used in standardized testing and psychological assessment to compare individual scores against a normative sample

Z-scores

• A z-score measures the distance of a score from the mean in units of standard deviation

• Z-scores allow researchers to compare scores from different distributions by converting them into a common scale

  • A z-score of 0 means the score is exactly at the mean

  • A positive z-score means the score is above the mean

  • A negative z-score means the score is below the mean

• Example:

  • If a test has a mean of 80 and a standard deviation of 8, a participant who scores 88 has a z-score of +1 — their score is one standard deviation above the mean

  • A participant who scores 72 has a z-score of −1 — their score is one standard deviation below the mean

Z-scores and percentile rank:

• Z-scores and percentile rank are directly related in a normal distribution:

  • A z-score of 0 corresponds to the 50th percentile

    • Exactly half of all scores fall below this point

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

    • The score is higher than approximately 98% of all scores in the distribution

  • Someone who scores at the 90th percentile has scored better than 90% of people who took the test

  • Someone who scores at the 38th percentile has scored better than only 38% of people who took the test

Regression toward the mean

• Regression toward the mean is the statistical phenomenon whereby extreme scores on an initial measurement tend to be closer to the mean when the same participants are measured again

• In other words, individuals who score unusually high or unusually low the first time they are measured tend to produce more average scores on a second measurement

  • This isn't because of any intervention, but simply due to chance variation

• As more data is collected, extreme scores become less influential and the overall distribution of scores moves closer to the true population mean

Why regression toward the mean matters

• It is a significant confounding variable in research

  • If a researcher selects participants based on extreme scores (e.g. those with very high anxiety) and then measures them again after an intervention, any improvement in scores may be due to regression toward the mean rather than the effect of the intervention

• Example:

  • A researcher selects participants who score extremely high on a stress scale and gives them a new relaxation program

  • When stress is measured again, scores are lower

  • This improvement may simply reflect regression toward the mean rather than the effectiveness of the program

• Researchers must account for regression toward the mean when interpreting results, particularly in studies that select participants based on extreme scores