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Measures of Central Tendency
What are measures of central tendency? This group of statistics is an attempt to describe a distribution of raw data by one score. This score is to be the most representative of the given data. We discussed three such measures in class: (1) Mean, (2) Median, and (3) Mode.

Mean. Mean is defined as the sum of all data points x divided by the number of data points n , or Mean = sum(x) / n

In the computation of mean, one is assuming equal intervals on the scale of a given measurement. Thus, the mean should only be employed if the scale of measurement is at least at the interval level of measurement.

Median. Median is a function of ordered data. That is, you have to first order your observations from the smallest to highest one according to the value. Then, the median is defined as the middle most score. In computation of the median, only the order is assumed. Thus, data has to be at least at the ordinal level of measurement.

Mode. Mode is defined as the most often occurring score. Since the computation of the mode does not assume either equal intervals, nor ordered observations, it can be used for all levels of measurement (including and mainly used with data at the nominal level of measurement).

Measures of Dispersion
Besides being able to describe the data by most representative score, one also wants to know how do the observations differ from one another. Why would one want to know this? Well, the dispersion (variability) of observations is associated with the measurement error and individual differences. Thus, the bigger the differences among observations, the more error exists in our measurement, or the more differences among the subjects. Additionally, the dispersion measures help to further discriminate one data set from another. For example, the measure of dispersion can be used to help in differentiating distributions that might have identical measures of central tendency. This is but one use of the measures of dispersion. Later, in hypothesis testing, the use of measures of dispersion will take on another meaning. In this lab, you will be introduced to four measures of dispersion: (1) Range, (2) Interquartile Range (IQR), (3) Standard Deviation, and (4) Variance.

Range. Range is defined as the difference between the upper real limit of the largest value and the lower real limit of the smallest value or

URL Xmax – LRL Xmin

Computation of range does assume order in your data set, since you have to identify the minimum and maximum value. With caution, range can be employed when data are at least at the ordinal level of measurement.

Interquartile Range. This range is defined as a difference between Q3 and Q1, where Q3 is the 75th percentile and Q1 is the 25th percentile. Xth percentile is a score that lies in an ordered data set at such a position that the X percent of data has value less or equal to this score.

IQR = Q3 – Q1

IQR has exactly the same requirements regarding the level of measurement as the range, above.

Variance and Standard Deviation. Variance for a population is defined as the sum of squared differences divided by the number of observations, or

Var = ∑(X – μ)2 / N

You can think of the variance as a squared distance that each score would have if all scores were to be equally distant from the mean. Standard Deviation is just a square root of variance. Standard Deviation can be thought of as an average distance that a data point in a given distribution has from the mean. Computation of Variance and Standard Deviation assumes equal intervals. Thus, data has to be at least at the interval level of measurement.

The Table below should help you figure out what graphs and measures of central tendency and dispersion are appropriate for different levels of measurement:

Table 1

Level of Measurement

Central Tendency

Dispersion

Graphs

Nominal

Mode

None

Bar chart

Ordinal

Mode, Median

Range, IQR

Bar chart, Histogram,

Interval

Mode, Median, Mean

Range, IQR, Variance, Standard Deviation

Histogram,

Ratio

Mode, Median, Mean

Range, IQR, Variance, Standard Deviation

Histogram

Look to Lab Assignment #1 to remind yourself of the level of measurement of each variable. Then, produce the appropriate measures of central tendency and dispersion for all four variables (see Table 1 for the appropriate measures). Round answers to the nearest 2 decimal places. (when appropriate)

Download the following file and open it in Microsoft Excel: Lab 1 – Excel Ver.xlsxՓաստաթղթի նախնական դիտում

Question 1.

Example Data (MPG))

Mean =
Median =
Mode =

Range
Standard Deviation =

Enter the appropriate scale of measure for this data set in the spaces below.

Scale of Measure =

Type of Graph =

Question 2.

Gender

HINT: You may not have to fill all the spaces provided. If an answer can not be provided, enter “none” in the space provided.

Mean =
Median =
Mode =

Range
Standard Deviation =

Enter the appropriate scale of measure for this data set in the spaces below.

Scale of Measure =

Type of Graph =

Question 3.

Current Salary

HINT: You may not have to fill all the spaces provided. If an answer can not be provided, enter “none” in the space provided.

Mean =
Median =
Mode =

Range
Standard Deviation =

Enter the appropriate scale of measure for this data set in the spaces below.

Scale of Measure =

Type of Graph =

Question 4.

Area Code

HINT: You may not have to fill all the spaces provided. If an answer can not be provided, enter “none” in the space provided.

Mean =
Median =
Mode =

Range
Standard Deviation =

Enter the appropriate scale of measure for this data set in the spaces below.

Scale of Measure =

Type of Graph =

Question 5.

Race

HINT: You may not have to fill all the spaces provided. If an answer can not be provided, enter “none” in the space provided.

Mean =
Median =
Mode =

Range
Standard Deviation =

Enter the appropriate scale of measure for this data set in the spaces below.

Scale of Measure =

Type of Graph =

Sample Solution

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