Statistics are often employed to compare one value to a group of values. In order to do this, it is necessary to describe the group of values first. One of the most useful quantities in descriptive statistics is a measure of central tendency. The “center” of a list of values gives the researcher an anchor around which, theoretically, the other members of the list occur. The “center” can be determined in several different ways; the most common are the mean, median, and mode (Berenson et al., 2012).
The mean, sometimes known as the average, is obtained by adding up all the values of the variable, then dividing by the number of values. This is the most accurate measure of central tendency, but only if it is computed for interval or ratio variables (Berenson et al., 2012). Both interval and ratio variables must have equal amounts between one number and the next. Also, ratio variables must have a true zero (Lane, 2011). An example of an interval variable is the time of day on a twelve hour clock. The quantity of time passed in one minute is the same whether it is between 12:01 and 12:02 or between 6:45 and 6:46. However, there is no true zero, because there is no point at which there is “zero time.” An example of a ratio variable is temperature measured using the Celsius or Kelvin scale (the Fahrenheit scale does not give a ratio variable because there is still heat or “temperature” even at 0° Fahrenheit). Both Celsius and Kelvin scales have a true zero at which there is no heat.

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The mean is not meaningful when it is computed for an ordinal or a nominal variable (Lane, 2011). Therefore, two other measures of central tendency can be used. The median is the middle value of a distribution, and it is particularly useful when describing ordinal variables (Berenson et al., 2012). An example of an ordinal variable is the score on a survey with answers from “strongly agree” to “strongly disagree.” The scores for a group of people can be ordered, so that one score is higher than another, but it is impossible to know the “sizes” of the intervals from one score to the next. The mode is the value of the variable that occurs most frequently in the distribution. It is used with nominal variables, in which there is no meaningful number or order of values, only names of values (Lane, 2011).

The median is determined by ordering the values, then taking the middle value. The mode is found by counting the number of times a given value appears in a distribution (Berenson et al., 2012). Theoretically, the three measures of central tendency will be equal when the distribution is normal (Lane, 2011). The following examples illustrate the calculation of mean, median, and mode.

1. Following are the resting pulses of subjects, reported in beats per minute. Calculate the mean resting pulse, rounded to the nearest beat. Be sure to show your work.
60  +  53  +  58  + 48  +  56  +  50  +  62  +  51  +  52 = 490
490/9 = mean = 54
The average resting pulse of this group of people was 54 beats/minute. This means that the other values centered around 54 — some were higher, some were lower.

2. A hospital kept track of the birth weights of babies measured in grams. The results are shown below.
2824    2973    3015    3025    3041    3102    3189
Find the median birth weight in grams. Provide an interpretation of your results.
The birth weights are already ordered, and the middle value is 3025 grams. (median =3025)
This means that half of the birth weights were above 3025 grams and half were below.

3. Last year, ten former employees of a company passed away. Their ages at death are listed below. The ages were reordered from youngest to oldest. Only 79 occurred more than once.
55   66 68   70 72 75 77 79 79 80
mode=79

In this case, the measure of central tendency did not give an accurate picture of the distribution, because 79 is actually very high. This is because the variable, age, is ratio level. Samples of interval and ratio variables often do not have a mode, and if they do have one, as in this case, it may not be very useful.

References
• Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Basic business statistics: Concepts and applications. Pearson Higher Education AU.
• Lane, D. (2011). Measures of Central Tendency. Online Statbook. Retrieved from http://onlinestatbook.com/2/summarizing_distributions/measures.html