# Central Tendency

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Measures of central tendency are scores that represent the center of the distribution. Three of the most common measures of central tendency are: –

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Mean Median Mode

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The Mean

The mean is the arithmetic average of the scores.

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Mean is the average of the scores in a distribution

_ X

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_________ i N

Σ Xi

Mean Example

Exam Scores 75 91 82 78 72 94 68 88 89 75

ΣX =sum all scores

n = total number of scores for the sample

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Pros

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Pros and cons of using mean

Summarizes data in a way that is easy to understand. Uses all the data Used in many statistical applications Affected by extreme values 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 20,000; 390,000 Mean = $44,167

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Cons

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E.g., average salary at a company

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Median

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The middle score of the distribution when all the scores have been ranked. If there are an even number of scores, the median is the average of the two middle scores.

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Central Tendency Example: Median

• 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891 • The median is the middle value when observations are ordered. – To find the middle, count in (N+1)/2 scores when observations are ordered lowest to highest.

• Median hotel rate:

– (35+1)/2 = 18 – 317

Median (con’t)

2 2 3 3 4 4 4 4 4 10 Number of Words Recalled in Performance Study

Pros and Cons of Median

• Pros

– Not influenced by extreme scores or skewed distributions. – Good with ordinal data. – Easier to compute than the mean.

• Cons

– May not exist in the data. – Doesn’t take actual values into account.

The mode. The mode is the score with the highest frequency of occurrences. It is the easiest score to spot in a distribution. It is the only way to express the central tendency of a nominal level variable.

Mode (con’t)

2 2 3 3 4 4 4 4 4 10 The mode is 4. Number of Words Recalled in Performance Study

M Mode (con’t)

72 81 87

72 83 88

73 85 90

76 85 91

78 86 92

This distribution is bimodal.

Demonstration

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Pros and Cons of the Mode

Pros

Cons

Good for nominal data. Good when there are two “typical” scores. Easiest to compute and understand. The score comes from the data set.

Ignores most of the information in a distribution. Small samples may not have a mode.

Scales of Measurement

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Nominal scale = mode Ordinal scale = median Interval(Discrete) scale = mean, median, or mode Ratio(Continuous) scale = mean, median, or mode

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What is dispersion? Explain two important measures of dispersion.

Measures of Dispersion

Why Study Dispersion?

An average, such as the mean or the median, only locates the centre of the data An average does not tell us anything about the spread of the data A small value for a measure of dispersion indicates that the data are clustered closely (the mean is therefore representative of the data) A large measure of dispersion indicates that the mean is not reliable (it is not representative of the data) 48 49 50 51 52 53

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Daily Computer Production

Daily Computer Production

WHAT IS DISPERSION?

Dispersion is the measure of the variation of the items. Measures of Dispersion are Range Quartile Deviation Mean Deviation Standard Deviation Variance

The Range

The simplest measure of dispersion is the range...

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