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Statistics
Calculating the Mean, Median, and Mode
Calculating the Range, Variance, and Standard Deviation
Chebyshev's Theorem and the Empirical Rule
Percentiles, Deciles, and Quartiles

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Calculating the Mean, Median, and Mode
The mean, median, and mode are single numbers that help describe how the individual scores in a data set are distributed in value. A data set consists of the observations for some variable is referred to as raw data or ungrouped data.

The Mean

The arithmetic mean is another name for the average of a set of scores. The mean can be found by dividing the sum of the scores by the number of scores.
For example, the mean of 5, 8, 2, and 1 can be found by first adding up the numbers. 5 + 8 + 2 + 1 = 16. The mean is then found by taking this sum and dividing it by the number of scores. Our data set 5, 8, 2, and 1 has 4 different numbers, hence the mean is 16 ÷ 4 = 4.

The Median

The median of a set of data values is the middle value once the data set has been arranged in order of its values. To find the mean of 2, 9, and 1, first arrange in order: 1, 2, 9. The median is the middle number or 2.

If you have an even number of values such as 1, 2, 5, and 8, the median is the average of the two middle numbers. The median for 1, 2, 6, and 8 is the average of 2 and 6 = 4.

The Mode

The mode of a set of data values is the number in the set that appears most frequently. For example, the number 5 appears three times in 1, 2, 5, 5, 5, 8, 8, 9. Since the number 5 appears the most times, it is the mode. A set of numbers that can have more than one mode, as long as the number appears more than once. In the data set 1, 2, 2, 3, 3, 4, 5. The mode is 2 and 3. We also can say that this data set is bimodal.

If no number appears more than once, then the data set has no mode.


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Calculating the Range, Variance, and Standard Deviation

The Range

Given a set of numbers, the range is equal to the maximum value in the data set minus the minimum value in the data set.

The range tells you how spread the entire data is. For example, given the numbers -3, 5, -9, and 19. The highest number is 19. The smallest number is -3. The range is therefore 19 - (-3) = 22.

Variance and Standard Deviation The variance and standard deviation of a data set measures the spread of the data about the mean of the data set.

The variance of a sample of size n represented by s2 is given by:
s2 =
[The sum of (x - mean)2]
(n-1)

The standard deviation can be calculated by taking the square root of the variance.

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Chebyshev's Theorem and the Empirical Rule
Chebyshev's Theorem

Chebysehev's theorem allows you to understand how the value of a standard deviation can be applied to any data set.

Theorem:  The fraction of any data set lying within k standard deviations of the mean is at least
1 -
1
k2

where k = a number greater than 1.
This theorem applies to all data sets, which include a sample or a population.

Empirical Rule

The empirical rule gives more precise information about a data set than the Chebyshev's Theorem, however it only applies to a data set that is bell-shaped.

Theorem:  
68% of the observations lie within one standard deviation of the mean.
95% of the observations lie within two standard deviations of the mean.
99.7% of the observations lie within three standard deviations of the mean.


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Percentiles, Deciles, and Quartiles
Given 45 out of 50 students had test scores less than 80. Since 45/50 = 90%. If you had a score of 80, you were in the 90th percentile.

The percentile for an observation x is found by dividing the number of observations less than x by the total number of observations and then multiplying this quantity by 100.

Once you can calculate Percentitles, you can also determine Deciles and Quartiles.

The First Quartile = the 25th Percentile
The Second Quartile = the 50th Percentile
The Third Quartile = the 75th Percentile


The First Decile = the 10th Percentile
The Second Decile = the 20th Percentile
... The Ninth Decile = the 90th Percentile

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