# Central Tendency

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• Published: January 20, 2013

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Central Tendency

Central Tendency Definition:
A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. The three most common measures of central tendency are the mean, the median, and the mode.

Measures of central tendency are measures of the location of the middle or the center of a distribution. The definition of "middle" or "center" is purposely left somewhat vague so that the term "central tendency" can refer to a wide variety of measures. The mean is the most commonly used measure of central tendency. The following measures of central tendency are discussed in this text:

1. Mean
2. Median
3. Mode
4. Trimean
5. Trimmed mean

Mean
Arithmetic Mean
The arithmetic mean is what is commonly called the average: When the word "mean" is used without a modifier, it can be assumed that it refers to the arithmetic mean. The mean is the sum of all the scores divided by the number of scores. The formula in summation notation is:

μ = ΣX/N

where μ is the population mean and N is the number of scores.

If the scores are from a sample, then the symbol M refers to the mean and N refers to the sample size. The formula for M is the same as the formula for μ.

M = ΣX/N

The mean is a good measure of central tendency for roughly symmetric distributions but can be misleading in skewed distributions since it can be greatly influenced by scores in the tail. Therefore, other statistics such as the median may be more informative for distributions such as reaction time or family income that are frequently very skewed

Click here for an interactive demonstration of properties of the mean and median.

The sum of squared deviations of scores from their mean is lower than their squared deviations from any other number.

For normal distributions, the mean is the most efficient and therefore the least subject to sample fluctuations of all measures of central tendency.

The formal definition of the arithmetic mean is µ = E[X] where μ is the population mean of the variable X and E[X] is the expected value of X. The geometric mean is the nth root of the product of the scores. Thus, the geometric mean of the scores: 1, 2, 3, and 10 is the fourth root of 1 x 2 x 3 x 10 which is the fourth root of 60 which equals 2.78. The formula can be written as:

Geometric mean =[pic]

where ΠX means to take the product of all the values of X. The geometric mean can also be computed by: 1. taking the logarithm of each number
2. computing the arithmetic mean of the logarithms
3. Raising the base used to take the logarithms to the arithmetic mean. The next page shows an example of this method using natural logarithms. |X |Ln(X) |

|1 |0 |
|2 |0.693147 |
|3 |1.098612 |
|10 |2.302585 |
|Geometric mean = 2.78 |Arithmetic mean = 1.024. |
| | |
| |EXP[1.024] = 2.78 |

The base of natural logarithms is 2.718. The expression: EXP [1.024] means that 2.718 is raised to the 1.024th power. Ln(X) is the natural log of X.

Naturally; you get the same result using logs base 10 as shown below.

|X |Log(X) |
|1 |0.0000 |
|2 |0.30103 |
|3 |0.47712 |
|10 |1.00000 |
|Geometric mean = 2.78 |Arithmetic mean = 0.44454. |
| | |
| |100.44454 = 2.78 |

If any one of the scores is zero then the geometric mean is zero. The geometric mean does not make sense if any scores...