STATISTICAL CONCEPTS IN ASSESSMENT OF LEARNING

Statistical Techniques allow to describe the performance of our students and make proper scientific inferences about their performance

A. Measures of Central Tendency

• Numerical values which describe the average or typical performance of given group in terms of certain attributes
• Basis in determining whether the group is performing better or poorer than the other groups

The Mean

• The mean is a single numerical measure of the typical or average performance of a group of students
• Is defined as the sum of observations divided by the number of observations

The Median

• The middlemost score
• Is unaffected by extreme examination scores
• is not necessarily one of the actual scores
• The median is the most appropriate average to calculate when the data result in skewed distributions

The Mode

• The most frequent score

B. Measures of Variability

• While measures of central tendency are useful statistics for summarizing the scores in a distribution, they are not sufficient. Two distributions may have identical means and medians, for example, yet be quite different in other ways. 

For example, consider these two distributions:

• GROUP A: 19, 20, 25, 32, 39
• GROUP B: 2, 3, 25, 30, 75
• Indicate or describe how spread the scores are
• The larger the measure of variability, the more spread the scores are, the group is said to be heterogeneous
• The smaller the measure of variability, the less spread the scores are, the group is said to be homogeneous

The Range

• Represents the distance between the highest and lowest scores in a distribution.
• Because it involves only the two most extreme scores in a distribution, the range is but a crude indication of variability. 

The Standard Deviation

• The most useful index of variability.
• It is a single number that represents the spread of a distribution
• Measure of average deviation or departure of the individual scores from the mean
• The more spread out scores are, the greater the deviation scores will be and hence the larger the standard deviation.
• The closer the scores are to the mean, the less spread out they are and hence the smaller the standard deviation. 

Quartile Deviation or Semi-interquartile Range

• Defined as one half the difference between quartile 3 (75th percentile) and quartile 1 (25th quartile) in a distribution
• Counterpart of the median
• Used when the distribution is skewed

B. Measures of Relative Position

Standard Scores

• use a common scale to indicate how an individual compares to other individuals in a group. 
• These scores are particularly helpful in comparing an individual’s relative position on different instruments.
• The two standard scores that are most frequently used in educational research are z scores and T scores.

z Scores

• the simplest form of standard score 
• expresses how far a raw score is from the mean in standard deviation units

T  Scores

• are z scores expressed in a different form. 
• To change a z score to a T score, simply multiply the z score by 10 and add 50.
• T = 10z + 50

Percentile Ranks

• A percentile in a set of numbers is a value below which a certain percentage of the numbers fall and above which the rest of the numbers fall.
• The median is the 50th percentile. 
• Other percentiles that are important are the 25th percentile, also known as the first quartile (Q1 ), and the 75th percentile, the third quartile (Q3 ). 
• To solve for percentile rank, add the number of students scoring below the value and the number of students scoring equal to the value divided by the total number of test takers.

Stanine Scores

• Tell the location of a raw score in a specific segment in a normal distribution which is divided into 9 segments
• Stanines 1, 2 and 3 reflect below average performance; 4, 5 and 6 reflect average performance; and 7, 8 and 9 reflect above average performance

SHAPES, DISTRIBUTIONS, and DISPERSION OF DATA

A. Shape

• Normal Distribution
• Rectangular Distribution
• U-Shaped Curve

B. Kurtosis

• Leptokurtic
• Mesokurtic
• Platykurtic

C. Unimodal, Bimodal and Multimodal Distributions of Test Scores

• Unimodal – one most common score
• Bimodal – two most common score
• Multimodal – more than two most common scores

D. Skewness

• Positively Skewed Distribution (mean > median > mode
• Negatively Skewed Distribution ( mode > median > mean)