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Statistics

Measures of Central Tendency

When we speak of data, we refer to either a sample or a population. We can use sample numerical descriptive measures to make inferences about the corresponding measures for a population. 

The central tendency of the set of measurements is the tendency of the data to cluster, or center, about certain numerical values. Let's describe some of these measurements of central tendency.

The mean of a set of quantitative data is the sum of the measurements, divided by the number of measurements contained in the data set.

The sample mean is denoted by \(\bar{x}\) and the population mean is denoted by \(\mu\).

Formula for a Sample Mean \(\overline{x}\)

\[\bar{x} = \frac{\sum\limits_{i=1}^{n} x_{i}}{n} \]

Example:

Calculate the sample mean of the following five measurements: 8, 4, 3, 9, 2. \[\bar{x} =\frac{\sum\limits_{i=1}^{5} x_{i}}{5} = \frac{8+4+3+9+2}{5} = 5.2\]

\[\bar{x} =\frac{\sum\limits_{i=1}^{5} x_{i}}{5} = \frac{8+4+3+9+2}{5} = 5.2\]

The median of a quantitative data set is the middle number when the measurements are arranged in ascending (or descending) order.

The sample median is denoted by M and the population median is denoted by \(\eta\).

Calculating a Sample Median

Arrange the measurements from the smallest to the largest.

  1. If n is odd, is the middle number.
  2. If n is even, M is the mean of the middle two numbers.

Example: Consider the following sample with the measurements: \(19, 5, 8, 22, 13, 21\)

To calculate the sample median, we first have to rearrange the measurements in order \(5,8,13,19,21,22\). Since there are an even number of measurements. We find the mean of the middle two measurements 13 and 19. \(\frac{13+19}{2}\). Therefore, \(M =16\).

The mode is the measurement that occurs most frequently in the data set.

Example: Ten participants rated their level of satisfaction on a 10-point scale on services delivered. The ten ratings were \[10,5,4,3,5,6,7,8,5,6\]

Since 5 occurs most often, the mode is 5.

For continuous data like that displayed by a histogram, the group with the highest frequency is called the modal class.

A data set is said to be skewed if one tail of the distribution has more extreme observations than the other tail. 

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