Statistical MethodsFrequency Distributions & Central Tendency by Adam J. McKee Using F. J. Gravetter and L. B. Wallnau's Essentials of Statistics for the Behavioral Sciences (4th Ed.). Introduction Data collection results in pages of numbers. These numbers must be organized before any sense can be made of them. This is the job of descriptive statistics—to simplify the organization and presentation of data. The Frequency Distribution A frequency distribution is an organized tabulation of the number of individual scores located in each category on the scale of measurement. The frequency distribution places the scores in order from highest to lowest. A Quiz Example The table to the right represents 20 quiz scores obtained for a 10 point quiz.
The highest score is x = 10 The lowest score is x = 4 The frequency for each score is located in the second column Frequencies can be used to find the total number of scores in the distribution Σf = N Getting ΣX From a Frequency Distribution Table When you need to perform calculations for scores that have been organized into a frequency distribution table, the safest procedure is to take the individual scores out of the table before you begin your computations. You can also multiply X times its frequency and sum the products Proportions and Percentages In general, the proportion associated with each score is: Because the proportion describes the frequency in relation to the total number (N), they are often called relative frequencies. Proportions are often converted to percents:
% = p x 100 Grouped Frequency Distributions The purpose of constructing a table is to obtain a simple, organized picture of the data. This can be accomplished by grouping the scores into intervals and then listing the intervals in the table instead of listing the individual scores. The result is called a grouped frequency distribution table. The Shape of Frequency Distributions
Shape Nearly all distributions can be classified as symmetrical or skewed. A symmetrical distribution is one where it is possible to draw a vertical line through the middle so that one side is mirror image of the other. Skew A skewed distribution is one where the scores pile up toward one end of the scale and taper off gradually at the other end. The section where the scores taper off toward one end of the distribution is called the tail. Positive and Negative Skew A skewed distribution with the tail on the right hand side is said to be positively skewed. A skewed distribution with the tail on the left side of the distribution is said to be negatively skewed. Central Tendency Central tendency is a statistical measure to determine a single score that defines the center of a distribution The goal is to find the score that is most typical or representative of the entire group; why we use averages for the final grade Mean The mean, AKA arithmetic average, is computed by adding all the scores in the distribution and dividing by the number of scores. The mean for a population is identified by the Greek letter mu, μ. The mean for a sample is identified by bar – x. Notation In general, we will use greek letters to identify population characteristics and letters from our alphabet to stand for sample values. Alternative ways of thinking about the mean Another way to think about the mean is to think of it as the amount everyone would get if the total were split up equally. The second way is to think of the mean as the balance point of the distribution The weighted Mean Often it is necessary to combine two sets of scores and then find the overall mean for the combined group. Note that the overall mean is not halfway between the original two sample means. The biggest group makes the largest contribution. Some Characteristics of the Mean Changing the value of any score will change the mean. Adding a new score or removing an old one will change the mean unless the score is exactly equal to the mean. Adding a constant to every score in the distribution has the effect of adding the same constant to the mean. Subtracting a constant from every score in the distribution has the effect of subtracting the constant from the mean. Multiplying of Dividing each score by a Constant If every score in a distribution is multiplied by a constant value, the mean will change in the same way. This method is used to change the unit of measurement (e.g., x 3 for changing yards to feet). The Median The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median. There is no statistical symbol for median—the word itself is used. Median when N is odd With an odd number of scores, you list the scores in order and the median is the middle score in the list 3,5,8,10,11: Median = 8 When N is an Even Number With an even number of scores, list the scores in order and then locate the median by averaging the middle two scores. 3,3,4,5,7,8 Median = (4+5)/2 = 9/2 = 4.5 A graph shows that the median divides the scores exactly in half. The Median Split Researchers who want to compare a high scoring group with a low scoring group will often split the group at the median. The Mode The mode is the score or category that has the greatest frequency. This is the appropriate measure for nominal scales. Say, religion, political affiliation, etc. It is possible that a distribution can have two or more scores with the same frequency– These are referred to as multimodal. If there are two scores, then it is bimodal. This page available at: |
