Applied Psychometrics: Statistical Concepts

• Applied Psychometrics focuses on the statistical concepts in measurement.

Learning Outcomes

• Review basic statistical concepts essential to psychometric theory and principles.
• Understand frequency distributions and measures of central tendency and variability.
• Understand the normal curve and the concept of divergence from normality.
• Define and differentiate between various types of standard scores.

Introduction

• Psychometricians design and use tests in various settings (business, academics, clinics, counselling).
• Measurement of individual performance on psychological tests is expressed as Scores.
• Scores are subjected to statistical methods for interpretation.
• Classifying and organizing scores is vital for deriving meaning.
• Statistics involves classifying, organizing, and analyzing data.
• Statistical methods help in understanding frequency distribution, central tendency, variability, normal distribution, skewness, kurtosis, p values, and statistical significance.
• Linear conversion of raw scores into standard scores is reviewed.

Frequency Distribution

• A distribution arranges test scores systematically for study.
• It facilitates the comparison of individual performances.
• Arranging test scores typically involves frequency distribution.
• Frequency distribution groups scores under subheads or classes, tabulating scores with their occurrence count.
• A grouped frequency distribution utilizes class intervals to represent actual test scores.
• The number and width of class intervals depend on the statistician's choice.
• Common grouping intervals are 3, 5, and 10 units in length.
• the higher class interval is called 'the upper limit' and the lower class interval is termed as “the lower limit' of the distribution.

Measures of Central Tendency

• Central tendency measures the 'average' or 'central position' score within a distribution.
• It describes group performance and facilitates comparison between groups.
• Three common measures: arithmetic mean (average), median, and mode.
• Arithmetic mean (average) is the sum of scores divided by the number of scores.
• Mean is a 'balance point' where negative deviations equal positive deviations and is sensitive to extreme scores.
• Median, the 'middle score', divides the distribution into two halves.
• It is responsive to the number of scores but less affected by extreme scores.
• For an odd number of scores, the median is the exact middle point.
• For an even number, it is the average of the two middle scores.
• Mode is the most frequent score, used with qualitative data, and can vary between samples.
• A distribution may have more than one mode.

Measures of Variability

• Variability measures how scores are scattered or clustered in a distribution.
• Distributions may have the same mean but differ in variability.
• Common measures: range, interquartile and semi-interquartile deviation, variance, and standard deviation.
• Range is the difference between the highest and lowest scores and is used as a rough measure.
• The interquartile range (Q) is the difference between the 75th (Q3) and 25th (Q1) quartile points.
• The semi-interquartile range is half of the interquartile range, corresponding closely to the median.
• Average deviation or variance is the mean of the deviations from the mean.
• Deviation scores are treated as positive, added, and divided by the number of scores; variance is resistant to sampling variation.
• Standard deviation is the square root of the variance and is widely used for descriptive and inferential statistics.

The Normal Curve

• Psychological test scores are often 'normally distributed'.
• is bell-shaped, smooth, symmetrical, unimodal, and continuous.
• It peaks at the center, tapering towards the 'tails'.
• Tails are asymptotic to the horizontal axis.
• The area under the curve is maximum in the middle, decreasing on both sides.
• Being symmetrical, mean, median, and mode are the same.
• 50% of scores are above the mean, and 50% are below.
• About 34% of scores occur between the mean and 1 standard deviation above or below the mean.
• About 68% of total scores fall within ±1 standard deviation.
• About 95% fall within ±2 standard deviations and about 99.7% within ±3 standard deviations.

Divergence from Normality: Skewness and Kurtosis

• Frequency distribution is defined by the presence or absence of symmetry.
• If symmetrical around the center, it is a normal distribution curve.
• Mean, median, and mode are equal in a symmetrical distribution.
• Skewness occurs when the mean and median are different, shifting the balance point.
• Positively skewed: extreme scores at the high end (Mean > Median > Mode).
• Negatively skewed: extreme scores at the low end (Mean < Median < Mode).
• Kurtosis describes the shape of a frequency distribution (peaked or flattened).
  • Leptokurtic: more peaked than normal.
  • Platykurtic: flatter than normal.
  • Mesokurtic: a normal flat curve.

P-Value and Statistical Significance

• A criterion determines if a statistically significant difference exists.
• 'Level of significance' reports how often a difference between groups would occur by chance.
• X equals 1 or 5, meaning a difference occurs by chance 1 out of 100 times (α = .01) or 5 out of 100 times (α = .05).
• p-values indicate the probability of a sample outcome if the null hypothesis is true.
• For example, a p-value of 0.06 means the sample results would occur by chance 6 times out of 100.

Standard Scores

• Needed to define performance by comparison and interpretation.
• Raw scores become interpretable by relating them to central tendency and variability.
• With psychological measures, scores are interpretable through conversion from one scale to another.
• A standard score relates the position of a raw score to other scores in a distribution.
• Standard scores indicate a testtaker's relative performance.
• Obtained by linear transformation, preserving proportionality.
• Standard scores can be derived through either linear or non-linear transformations.
• Linear transformations preserve interscore distance.
• Examples include z score and T score.
• Non-linear transformations are used when raw scores are not normally distributed.
• Normalizing involves stretching the skewed curve into a normal curve.
• Resulting in a normalized standard score scale, for example percentile rank and stanine scores.

Z-Score

• Represents how many standard deviations a raw score falls above or below the mean.
• Mean and standard deviation of a set of z scores are 0 and 1, respectively.
• Formula: z = (X - M) / SD, where X is the raw score, M is the mean, and SD is the standard deviation.
• Properties:
  • The mean of a set of z-scores is always 0.
  • The SD of a set of standardized scores is always 1.
  • The distribution curve is the same as unstandardized scores.
  • Scores can be positive or negative.

T Score

• Another standardized score with a mean of 50 and an SD of 10.
• The scale is called a "fifty plus or minus ten" scale.
• Formula: T = 10z + 50.
• T-score can never be negative which is an advantage over z-score.

Stanine scores

• Stanine scores have a mean of 5 and an SD of approximately 2.
• 'Stanine' comes from 'standard' and 'nine', as it divides SD into nine units.
• It has values ranging from 1 to 9, and is simple because it requires only a single digit.
• Stanine is a coarse unit because the difference between successive stanines values is one half of a standard deviation.

Percentile Ranks

• Describe the location of a raw score in relation to other scores in a distribution.
• Percentile rank has directness of meaning.
• For example, a percentile rank of 68 means a student performed better than 68% of peers.
• Percentile ranks are easy to understand, even without statistical training.

Summary

• Statistical tools are the basis for making test scores interpretable.
• Frequency distribution arranges responses to show the number of observations for each category.
• Adequate description requires measures of central tendency and information about variability and shape.
• Variability measures how scores are scattered or clustered.
• The normal curve is symmetrical, continuous, and asymptotic to the horizontal axis.
• It describes the frequency of variables with accuracy.
• Divergence from normalcy is determined by skewness and kurtosis.
• A derived score with meaning and interpretation after derivation.
• Different systems for standard scores exist.
• The four main types are z scores, T scores, stanines, and percentile ranks.
• Knowledge of statistics is essential for communicating the meaning of data.