Measures of Central tendency and Variability

            Central tendency is a statistical measure that identifies a single score as representative of an entire distribution of scores. The goal of central tendency is to find the single score that is most typical or most representative of the entire distribution. Unfortunately, there is no single, standard procedure for determining central tendency. The problem is that there is no single measure that will always produce a central, representative value in every situation. There are three main measures of central tendency: the arithmetical mean, the median and the mode. 

            The mean of a set of scores (abbreviated M) is the most common and useful measure of central tendency.  The mean is the sum of the scores divided by the total number of scores. The mean is commonly known as the arithmetic average. The mean can only be used for variables at the interval or ratio levels of measurement. The mean of [2 6 2 10] is (2 + 6 + 2 + 10)/4 = 20/4 = 5. One can think of the mean as the balance point of a distribution (the center of gravity). It balances the distances of observations to the mean. Another measure of central tendency is the median, which is defined as the middle value when the numbers are arranged in increasing or decreasing order. The median is the score that divides the distribution of scores exactly in half. The median is also the 50th percentile. The median can be used for variables at the ordinal, interval or ratio levels of measurement. If for example, daily expenses are $50, $100, $150, $350, $350 the middle value is $150, and therefore, $150 is the median. For odd number of count the median is middle value. If there is an even number of items in a set, the median is the average of the two middle values. For example, if we had four values—$50, $100, $150, $350—the median would be the average of the two middle values, $100 and $150; thus, 125 is the median in that case. The median may sometimes be a better indicator of central tendency than the mean, especially when there are extreme values. Another indicator of central tendency is the mode, or the value that occurs most often in a set of numbers. In other words, the mode is the score or category of scores in a frequency distribution that has the greatest frequency. In the set of expenses mentioned above, the mode would be $350 because it appears twice and the other values appear only once. The mode can be used for variables at any level of measurement (nominal, ordinal, interval or ratio). Sometimes a distribution has more than one mode. Such a distribution is called multimodal. A distribution with two modes is called bimodal. Note that the modes do not have to have the same frequencies. The tallest peak is called the major mode; other peaks are called minor modes. Some distributions do not have modes. A rectangular distribution has no mode. Some distributions have many peaks and valleys.

              Variability provides a quantitative measure of the degree to which scores in a distribution are spread out. The greater the difference between scores, the more spread out the distribution is.  The more tightly the scores group together, the less variability there is in the distribution. Variability is the essence of statistics. The most frequently used methods of measurement of this variance are: range, deviation and variance, interquartile range and standard deviation. The range is simply the difference between the highest score and the lowest score in a distribution plus one. This statistic can be calculated for measurements that are on an interval scale or above. In dataset with 10 numbers {99,45,23,67,45,91,82,78,62,51}, the highest number is 99 and the lowest number is 23, so 99−23=76; the range is 76. The interquartile range (IQR) is a range that contains the middle 50% of the scores in a distribution. It is computed as follows: IQR=75th percentile−25th percentile. A related measure of variability is called the semi-interquartile range. The semi-interquartile range is defined simply as the interquartile range divided by 2. Variance can be defined as a measure of how close the scores in the distribution are to the middle of the distribution. Using the mean as the measure of the middle of the distribution, the variance is defined as the average squared difference of the scores from the mean. When the scores are spread out or heterogeneous, the measure of variability should be large. When the scores are homogeneous the variability should be smaller. Another measure of variability is the standard deviation. The standard deviation is simply the square root of the variance. The standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal (see Probability) because the proportion of the distribution within a given number of standard deviations from the mean can be calculated. Therefore standard deviation is the average distance from the mean. So the mean is the representative value, and the standard deviation is the representative distance of any one point in the distribution from the mean.

              While the measures of central tendency convey information about the commonalties of measured properties, the measures of variability quantify the degree to which they differ. If not all values of data are the same, they differ and variability exists. The measures of central tendency should be complemented by measures of variability for the same reason.

The Research process: an independent and dependent variables

 

        Research is the foundation of any science, including both hard sciences such as physics, chemistry and the social sciences such as psychology, management and education. The steps and process involved in the research can vary depending on the type of research being done and the hypothesis being tested. Research methods such as Naturalistic observation and surveys are often less structured, where as experimental methods are more structured. Depending upon what is observed or experienced, new theories are developed. There are aspects of a theory or aspects of a study that can change or vary as part of interaction within the theory, defined as variables. Variables are anything that can change of effect the results of a study. In an experimental method, the experiment is conducted by changing the value of one variable and measuring the changes in another variable while holding or assuming surroundings constant. There is no limit to the number of variables that can be measured, although the more variables, the more complex the study and the more complex the statistical analysis.

        Every experiment has at least two types of variables: an independent and dependent. An independent variable is the variable that the researchers systematically manipulate in the experiment. An independent variable is measured, manipulated, or selected by the experimenter to determine its relationship to an observed phenomenon. This might be a variable that you control, like a treatment, or a variable not under your control, like an exposure. It also might represent a demographic factor like age or gender. While the independent variable is often manipulated by the researcher, it can also be a classification where subjects are assigned to groups. In a study where one variable causes the other, the independent variable is the cause. In a study where groups are being compared, the independent variable is the group classification. In a research to demonstrate the increasing alcohol consumption during pregnancy actually causes a reduction in birth weight, researchers randomly assigned 50 pregnant rat either to an experimental group or to a control group. The 25 rats in the experimental group had bottles filled in with mixture of pure water and alcohol, 25 rats in the control group had bottles with water. Other than this, researchers treated all rats as much alike as possible. Aim was to have the two groups differ systematically along only one variable: alcohol versus pure water. This is independent variable; idea was to manipulate this variable independently of other factors such as diet that might affect pregnancy. In this example, birth weight is the dependent variable that is it represents the outcome that we measure, an outcome that is dependent on the manipulation of the independent variable. With experiments, then, researchers systematically manipulate the independent variable to determine if it causes a difference in the dependent variable. There are three ways to manipulate independent variables: presence or absence technique, amount technique and type technique. In presence or absence technique, the independent variable can be manipulated can be manipulated by presenting a condition or treatment to one group of individuals and withholding the condition or treatment from another group of individuals. In amount technique, the independent variable can be manipulated by varying the amount of a condition or variable such as the amount of a drug which is given to children within a learning disorder. In type technique, the independent variable is to vary the type of the condition or treatment administered.  

        In an experiment, a dependent variable is the factor which is observed and measured to determine the effect of the independent variable, that is, that factor that appears, disappears, or varies as the experimenter introduces, removes, or varies the independent variable. The dependent variable is the participant’s response. The dependent variable is the outcome of experiment. In an experiment, it may be what was caused of what changed as a result of the study, in a comparison of groups; it is what they differ on. In research study on rats, described in previous paragraph, the experimental group is exposed to alcohol, and the control group is not. If we observe that the rat pups in the groups differ reliably in birth weight, and then we can conclude alcohol exposure caused this difference. In this example birth weight is a dependent variable.

 

 

 

        In an experiment, the independent variable is the variable that is varied or manipulated by the researcher, and the dependent variable is the response that is measured. An independent variable is the presumed cause, whereas the dependent variable is the presumed effect. The independent variable is the antecedent, whereas the dependent variable is the consequent. In experiments, the independent variable is the variable that is controlled and manipulated by the experimenter; whereas the dependent variable is not manipulated, instead the dependent variable is observed or measured for variation as a presumed result of the variation in the dependent variable. Dependent variables can be influence by controlled variables.