DESCRIPTIVE STATISTICS REPORT

The Descriptive Statistics Report provides the statistical characteristics of the test scores for a class. All the information that has been drawn from the list of scores is available, but what the scores show is hard to understand from the complete list, especially if the number of scores is large. Summary statistics provide a more comprehensible picture. The distribution statistics report is available for part scores as well as for the total score. The part scores will be reflected Descriptive Statistics Report or the Score Distribution Table.

Two characteristics of the distribution are of primary interest:

Central Tendency refers to the score that is most representative of the entire distribution.

Variability refers to the tendency of the scores to be either spread out or bunched up around the center.

Two characteristics of the distribution are of primary interest:

CENTRAL TENDENCY

Three measures of central tendency are provided.

  1. Mean is the arithmetic average, the sum of all scores divided by the total number of scores. If the units on the score scale are equal, which is true of most tests, and if the distribution of scores is symmetrical, the mean is generally the best measure of central tendency. However, extreme scores heavily influence the mean. Consequently, if the distribution is skewed (that is, if there are extreme scores at one end that are not balanced by extreme scores at the other end of the distribution), the mean may not adequately represent the center of the distribution.
  2. Median (50th Percentile) is the point on the score scale where half the cases lay either above or below. Because extreme scores do not influence the median, it represents the central tendency of skewed distributions better than the mean.
  3. Mode is the most frequently appearing score. It may be thought of as the most typical score. However, the same frequency may occur at more than one score, so there can be more than one mode. The mode is indicated on the Overall Score Distribution Table.

MEASURES OF VARIABILITY

Variance is the average squared deviation around the mean. The mean is subtracted from each score; the resulting difference is squared; and the sum of the squared differences is divided by the total number of scores.

Standard Deviation is the square root of the variance, and it represents a distance on the score scale. If the test scores are distributed as a normal curve, 68 percent of the scores are between one standard deviation below the mean and one standard deviation above the mean while 98 percent of the scores are between two standard deviations below the mean and two standard deviations above the mean.

Range of Scores is the distance on the score scale that includes all the scores, calculated by adding 1 to the difference between the highest and lowest scores. Because the range depends on the values of just two scores, it is not a reliable, stable statistic.

The distribution statistics report provides other statistics from which other measures of variability can be obtained: the 10th, 25th, 75th, and 90th Percentile Scores, which are the points on the score scale that divide the distribution of scores into the specified percentages. The inter quartile range is the distance between the 25th and 75th percentiles. By definition, 50% of the scores are included in the interquartile range. The semi-interquartile range, which is half the interquartile range, is often used with the median, just as the standard deviation is used with the mean in order to describe the variability and central tendency of a distribution.

OTHER INFORMATION

Other items on the report include the Group Size (total number of scores), the sum of scores and the sum of squared scores, the highest and lowest obtained scores, the highest possible score and the percentages that the mean is of the highest possible score and the highest obtained score. The last two measures indicate the difficulty of the test. If the percentages are high (e.g., 80 or above), the test was easy for this group. If the percentages are low (e.g., below 50), the test was hard for this group. The highest possible score and the percent that the mean is of the highest possible score are not printed on part-score reports because they cannot always be determined.

OVERALL TOTAL SCORE DISTRIBUTION TABLE

The distribution of scores is the first level of score summarization. The distribution is presented in several ways:

Frequency is the number of test-takers with each score.

Cumulative Frequency is the number of test-takers at or below each score.

Percent is the percent of test-takers that have each score.

Cumulative Percent is the percent of test-takers at or below each score.

The Percentile Rank and Standard Score corresponding to each raw score are also shown. Percentile rank is the percent of test-takers who scored below the mid-point of a given score. Standard score, also known as a z-score, is computed by subtracting the population mean of the individual raw score and dividing the remainder by the standard deviation resulting in scores that have a mean of 50 and a standard deviation of 10. Converting raw scores to standard scores is most helpful if you want to compare or combine the results from two or more tests. By converting to standard scores, you are converting two or more tests (with each having different distribution properties (e.g., mean, standard deviation) to the same scale, one that has a mean of 50 and a standard deviation of 10, regardless of what the original raw score descriptive statistics were.

Frequency Distribution of test scores that provides a graphical representation of the data is included in all reports. Examination of the frequency distribution can give a quick indication of whether or not the scores approximate a normal distribution, are skewed or exhibit other abnormalities.