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Statistics Study Guide

Mean, Median, Mode, T and Z scores

The purpose of this study guide is to help students better understand the mean, median, mode, Z-score standard deviation. The study guide includes material posted on different web sources.


What is a mean? In statistics, the mean is the mathematical average of a set of numbers. The average is calculated by adding up two or more scores and dividing the total by the number of scores.

Consider the following number set: 2, 4, 6, 9, 12. The average is calculated in the following manner: 2 + 4 + 6 + 9 + 12 = 33 / 5 = 6.6. So the average of the number set is 6.6.

More on mean definition here and here


What is a median? The median is the middle number (in a sorted list of numbers).

To find the Median, place the numbers you are given in value order and find the middle number.
Example: find the Median of {13, 23, 11, 16, 15, 10, 26}.
Put them in order: {10, 11, 13, 15, 16, 23, 26}
The middle number is 15, so the median is 15.

What is the median? Median in statistics is the same thing. Well, it’s not a grassy area, but the median does refer to the number in the middle of a data set. When finding the median of a data set, you have to make sure that the numbers are put in order first. If there is an odd number of data in the list, there is only one number that is exactly in the middle of the data. But if there is an even number of data points, then there are two numbers in the middle. In that case, you have to add those two numbers together and then divide by two to find the median.


What is a mode ? The mode of a data set refers to the number that occurs most often. If there is not a number that occurs more than any other, we say there is no mode for the data. It is possible to have more than one mode for a data set.

More examples and more practice on mean, median, and mode is here

T and Z scores

Figure1 : T and Z scores are based on the statistical unit of the standard deviation. Shown here is the classical bell-shaped curve with the percent of a population lower than that value shown next to the curve.

The T-score is the number of standard deviations below the average for a young adult at peak bone density. There are different T-scores depending on which group of young adults was used as the reference (for example, Caucasian women, and Hispanic men). The Z-score is the number of standard deviations below an average person of the same age. There are also different Z-scores depending on the group used as a reference (for example, the group could include everybody of the same age, or it could be limited to people with the same age, race, gender and weight). Furthermore, a person can have one T-score at the femoral neck, another at the total hip, and another at the spine.

More on T and Z score is here (more details about the relationships between T-score, Z-score and g/cm2, including conversion formulas, data tables and some of the misconceptions about the interpretation of the scores.)

What are NCE Scores? Normal Curve Equivalent (NCE)

Normal Curve Equivalent scores are similar to Percentile Ranks, but they are based on an equal interval scale. This means that the difference between any two successive scores on the NCE scale has the same meaning throughout the scale. NCEs range from 1 to 99. They are useful for making meaningful comparisons between different achievement tests and for statistical computations, such as determining an average score for a group of students. NCEs are mostly used for research purposes and government program evaluations. The NCE is included on these reports: the Diagnostic Report, the Growth Report, the Summary Report, and the Test Record Report. And more , Basic Statistics and Test Score Interpretation

Understanding Statistics Tutorial - ECS

Grade Equivalence

If a student obtains a score on a test that is equal to the median score for all the beginning sixth-graders (September testing) in the normal group, then that student is given a grade equivalent of 6.0. If a student obtains a score equal to the median score of all beginning fifth-graders, he is given a grade equivalent of 5.0. If a student should score between these two points, linear interpolation would be used to determine his grade equivalent. Because most school years run for ten month, successive month are expressed as decimal. Thus, 5.1 would refer to the average performance of fifth-grader in October, 5.2 in November, and so on to 5.9 in June.

GE limitations:

  • Problem of interpolation
  • Gives very little information about the percentile standing on a person within his class
  • Contrary to what the number indicates
  • Norms are not standards

Stanine Score

What is Stanine Score? Definition: a nine-point scale used for normalized test scores, with 1-3 below average, 4-6 average, and 7-9 above average; one of the steps in a nine-point scale of standard scores and more .

Understanding and Explaining Standardized Test Scores

Interpreting Stanine Score

Standard Error of Measurement

(SEM) is the standard deviation of errors of measurement that are associated with test scores from a particular group of examinees. When used to calculate confidence bands around obtained test scores, it can be helpful in expressing the unreliability of individual test scores in an understandable way. Sore bands can also be used to interpret intra-individual and inter-individual score difference. Interpreters should be wary of over-interpretation when using approximation for correctly calculated score bands. It is recommended that SEMs at various score levels be used in calculating score bands rather than a single SEM value. And more is here


Introduction to statistics videos

More Practice Exercises:

Mean, Median, and Mode. Passing the test

Mean, Median, and Mode, Inserting the Missing Data

Interpreting Scores, more practice for final exam

z and t Scores, more practice for finals

More Practice on Finding Standard Deviation, final exam preparation

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