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**Measures of Dispersion**

Variance and Standard Deviation

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**Basic Assumptions about Distributions**

We should be able to plot the number of times a specific value occurs on a graph using a line chart or histogram (interval/ratio data) Some distributions will be normal or bell-shaped. Some distributions will be bi-modal or will have data points distributed irregularly. Some distributions will be skewed to the right or skewed to the left. Theoretically, samples taken from one population, should over time, approximate a normal distribution. We should have a normal distribution if we are to use inferential statistics.

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**Other reasons to use Measures of Dispersion**

To see if variables taken from two or more samples are similar to one another. To see if a variable taken from a sample is similar to the same variable taken from a population – in other words is our sample representative of people in the population at least on that one variable.

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**Variation in Two Samples**

1 2 3 4 5 7 9 6 10 Mo = 4 Mo, 3, 9 Mdn = 4 Mdn = 6 Mean = 4 Mean = 6

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Sample 2

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Normal Distributions are Bell-shaped and have the same number of measures on either side of the mean. Note: According to Montcalm & Royse only unimodal distributions can be normal distributions.

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**Normal Distributions 50% of all scores are on either side of the mean.**

The distribution is symmetrical – same number of scores fall above and below the mean. The mean is the midpoint of the distribution. Mean = median = mode The entire area under the bell-shaped curve = 100%.

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**A standard deviation is:**

The degree to which each of the scores in a distribution vary from the mean. (x – mean) Calculated by squaring the deviation of each score from the mean. Based on first calculating a statistic called the variance.

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Formulas are: Variance = Sum of each deviation squared divided by (n -1) where n is the number of values in the distribution. Standard Deviation = the square root of the sum of squares divided by (n – 1).

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**Using Sample 1 as an example**

(1-4) = -3 9 Mean = 4 2 (2-4) = -2 4 3 (3-4) = -1 (4-4) = 0 Variance S.D. 28/(8-1) Sq Root 4 5 (5-4) = 1 6 (6 - 4) = 2 7 (7 = 4) = 3 Total 28

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**Another variance/SD example**

1 -5.00 25.00 Mean = 6 2 -4.00 16.00 4 -2.00 4.00 8 2.00 10 Variance = 90/(6-1) SD = sq root of 18 11 5.00 18.00 4.24 Total 0.00 90.00

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**Other Important Terms in This Chapter**

Mean squares – the average of squared deviations from the mean in a set of numbers. (Same as variance) Interquartile range – points in a set of numbers that occur between 75% of the scores and 25% of the scores – that is, where the middle 50% of all scores lie (use cumulative percentages) Box plot – gives graphic information about minimum, maximum, and quartile scores in a distribution.

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Box Plot Line in middle is median. Upper line is maximum; lower line is minimum. Red box is the interquartile range.

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**Interquartile Range Test Scores Frequency Percent Cumulative 100 3 25%**

100% 90 75% 80 50% 70 1 8.3% 60 2 16.7% Total 12 100.0%

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**This information is important to our discussion of normal distributions**

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**Central Limit Theorem (we will discuss this in two weeks) specifies that:**

50% of all scores in a normal distribution are on either side of the mean. 68.25% of all scores are one standard deviation from the mean. 95.44% of all scores are two standard deviations from the mean. 99.74% of all scores in a normal distribution are within 3 standard deviations of the mean.

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**Therefore, we will be able to**

Predict what scores are contained within one, two, or three standard deviations from the mean in a normal distribution. Compare the distribution of scores in samples. Compare the distribution of scores from populations to samples.

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**To calculate measures of central tendency and dispersion in SPSS**

Select descriptive statistics Select descriptives Select your variables Select options (mean, sd, etc.)

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SPSS output

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