Assessment, Statistics, and Research

The z- score defined  A z -score indicates how far a raw score lies from the arithmetic mean of a normally distributed set of scores. It expresses this distance in standard‑deviation units. Thus, a z -score of 1.0 means the score is one standard deviation above the mean, while a z -score of –1.0 means it is one standard deviation below the mean. z-scores are typically plotted along the x‑axis of a normal distribution —the familiar bell‑shaped curve. CALCULATE z -scores by using the calculator at the bottom of this page. Here's an image of the normal curve with z -scores below the curve. In APA style, z -scores are reported in lowercase italics . The uppercase Z refers to a different statistic. A z-score is calculated by subtracting the mean (M) from the raw score (X) and dividing the result by the standard deviation (SD): z = ( X - M ) / SD Example: If a test score is 60, the mean is 50, and the standard deviation is 10, then:      [ 60 - 50 = 10 ]     ...

 Kurtosis is a statistical concept. The value indicates whether a distribution is similar to the normal curve or different from the normal curve. Compared to the normal curve, kurtotic distributions of data appear either peaked in the middle or flat. In a normal distribution, the value of kurtosis = 0. The peaked distribution has a positive value. It's called leptokurtic (think leap). The flatter distribution has a negative value. It's called platykurtic (think of the animal, Platypus). There are different formulas for calculating kurtosis. In Excel, the function for kurtosis can be found under Formulas, More Functions. In the drop down list, choose KURT. Please check out my website     www.suttong.com    and see my books on    AMAZON         or   GOOGLE STORE Also, consider connecting with me on     FACEBOOK     Geoff W. Sutton         TWITTER  ...

Understanding the Bell Curve and Standard Deviation The bell curve—also known as the normal curve or normal distribution—has mathematical properties that allow researchers to understand where scores (or data points) fall in relation to one another. In a normal distribution, most scores cluster around the center, with fewer scores appearing as you move toward the extremes. Central Tendency in a Normal Curve In a normal distribution, the three measures of central tendency—mode, median, and mean—all fall at the same central point. This midpoint divides the distribution into two equal halves. Standard Deviation Units On the x‑axis of a normal curve, the mean is placed at zero. Standard deviation units extend outward in both directions from the mean. These units help us understand how far a score is from the average. The height of the curve at any point represents the percentage of scores in that area. A large portion of scores fall close to the mean. In fact, about 68% of all scores lie be...

  Skewed Distributions* Skewed distributions have one tail that is longer than the other tail compared to the "normal" distribution, which is perfectly symmetrical. Skew affects the location of the central values of the mean and median. Positive Skew Below is an image of positive skew, which is also called right skew. Skew is named for the "tail." If you had statistics, you may have heard a professor say, "the tail tells the tale." The tail is the extended part of the distribution close to the horizontal axis. The large "hump" area to the left represents the location of most data. In behavioural science, the high part often refers to the location of most of the scores. Thus, in positively skewed distributions, most of the participants earned low scores and few obtained high scores as you can see by the low level of the curve, or the tail, to the right. Negative Skew As you might expect, negatively skewed distributions have the long tail on the le...

In testing, measurement error usually refers to the fact that the same people can obtain different scores on the same test at different times. In a broad sense, measurement error can also refer to the degree of accuracy of a test to correctly identify a condition, which is discussed as test validity. Recall that test score reliability is a necessary but insufficient condition for test score validity. Many tests in psychology, medicine, and education are useful. The reliability of the scores will vary depending on such factors as the properties of the test itself as well as how well the user follows standard procedures in administering the test, environmental factors that can affect the scores, and factors within the person taking the test. The scores on many tests conform to the pattern called the normal curve or bell curve. In classical test theory, the scores people obtain on tests are simply called obtained scores (symbol X). Statisticians consider the variation in scores t...