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Descriptive statistics are used to illustrate the distribution of a variable or variables in a sample. Their purpose is to summarize data in a simple and understandable way. They are typically used only for describing the data rather than testing for significance and describe the central tendency and the dispersion of data. Measures of central tendency — mean, median, and mode — attempt to provide a snapshot of the center of a distribution. Measures of dispersion — range, variance, and standard deviation — attempt to provide a snapshot of how a distribution of the observed scores of a variable varies around the mean.
Before we begin to understand the measures of central tendency and dispersion, it is important to understand a distribution and an array. The distribution of a variable is the value of each individual score or category (for example: 70, 35, 32, 18, 45, 55, 43, 55, 17 could be the ages of individuals participating in a survey). An array is when these scores are sorted in an ascending manner (example: 17, 18, 32, 35, 43, 45, 55, 55, 70).
Measures Of Central Tendency
The mode is the most frequently occurring score in a distribution. In the above example, two participants are of the age 55, making it the mode. The median is the exact center of an array of variables.
In this example, the ages of four participants are below 43 and the ages of four participants are above 43. This makes 43 the median age of this sample. When mathematically calculating the median for a sample where there are an even number of participants, the median is the two middle scores in an array divided by two. The mean is the average score in the distribution of a variable. In this example, adding up all of the ages of the participants and then dividing it by the number of participants or the sample size n = 9 (example: 17 + 18 + 32 + 35 + 43 + 45 + 55 + 55 + 70 = 370, now divide 370 by n = 9 to get the mean age of this sample: 41.11 years).
Measures Of Dispersion
The standard deviation is considered an important measure of spread. This is because, if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. In a normal distribution, about 68 percent of the scores are within one standard deviation of the mean and about 95 percent of the scores are within two standard deviations of the mean.
- Healey, J. (2009) Statistics: A Tool for Social Research, 8th edn. Wadsworth, New York.
- Larson, R. & Farber, E. (2005) Elementary Statistics: Picturing the World, 3rd Prentice Hall, Englewood Cliffs, NJ.