## Range

The variety is the easiest measure the variation come find. It is simply the greatest value minus thelowest value. Range = maximum - MINIMUMSince the variety only offers the largest and also smallest values, it is greatly impacted by too much values,that is - it is no resistant to change.

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## Variance

"Average Deviation"The variety only involves the smallest and also largest numbers, and also it would certainly be desirable to have actually astatistic which involved all of the data values.The an initial attempt one can make at this is something castle might contact the typical deviation indigenous mean and define it as: The difficulty is that this summation is constantly zero. So, the typical deviation will always be zero. The is why the median deviation is never ever used.Population VarianceSo, to keep it from being zero, the deviation from the median is squared and called the "squareddeviation indigenous the mean". This "average squared deviation indigenous the mean" is dubbed the variance. Unbiased estimate of the populace VarianceOne would expect the sample variance to simply be the population variance v the populationmean changed by the sample mean. However, one of the major uses of statistics is to calculation thecorresponding parameter. This formula has the difficulty that the estimated value isn"t the same asthe parameter. To against this, the sum of the squares of the deviations is divided by one lessthan the sample size. ## Standard Deviation

There is a trouble with variances. Recall the the deviations to be squared. That means that theunits were additionally squared. To gain the units ago the very same as the initial data values, the squareroot should be taken.  The sample traditional deviation is not the unbiased estimator for the populace standard deviation.The calculator does not have actually a variance vital on it. It does have a standard deviation key. Youwill need to square the typical deviation to uncover the variance.

## Sum of Squares (shortcuts)

The sum of the squares of the deviations indigenous the method is offered a faster way notation and also severalalternative formulas. A little algebraic simplification returns: What"s wrong with the first formula, friend ask? think about the following instance - the last heat arethe totals for the columns complete the data values: 23 division by the number of values to gain the mean: 23/5 = 4.6 Subtract the median from each value to acquire the number in the 2nd column. Square each number in the 2nd column to acquire the worths in the 3rd column. Total the number in the third column: 5.2 divide this full by one less than the sample dimension to acquire the variance: 5.2 / 4 = 1.3
 x  4 4 - 4.6 = -0.6 ( - 0.6 )^2 = 0.36 5 5 - 4.6 = 0.4 ( 0.4 ) ^2 = 0.16 3 3 - 4.6 = -1.6 ( - 1.6 )^2 = 2.56 6 6 - 4.6 = 1.4 ( 1.4 )^2 = 1.96 5 5 - 4.6 = 0.4 ( 0.4 )^2 = 0.16 23 0.00 (Always) 5.2
Not too bad, girlfriend think. But this can get pretty poor if the sample typical doesn"t happen to it is in an"nice" reasonable number. Think about having a typical of 19/7 = 2.714285714285... Thosesubtractions gain nasty, and also when girlfriend square them, they"re really bad. One more problem through thefirst formula is the it needs you to know the typical ahead of time. For a calculator, this wouldmean that you need to save all of the numbers that were entered. The TI-82 walk this, yet mostscientific calculators don"t.Now, let"s take into consideration the faster way formula. The only things the you require to uncover are the amount of thevalues and also the amount of the worths squared. There is no subtraction and no decimals or fractionsuntil the end. The last row consists of the sums of the columns, just like before. Document each number in the very first column and also the square of each number in the second column. Total the first column: 23 total the 2nd column: 111 Compute the amount of squares: 111 - 23*23/5 = 111 - 105.8 = 5.2 divide the sum of squares by one less than the sample dimension to acquire the variance = 5.2 / 4 = 1.3
 x x^2 4 16 5 25 3 9 6 36 5 25 23 111

## Chebyshev"s Theorem

The relationship of the values that loss within k conventional deviations the the average will be in ~ least , whereby k is one number greater than 1."Within k conventional deviations" interprets together the interval: to .Chebyshev"s theorem is true for any sample set, not matter what the distribution.

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## Empirical Rule

The empirical dominion is just valid for bell-shaped (normal) distributions. The complying with statementsare true. Around 68% that the data values loss within one traditional deviation that the mean. About 95% the the data values fall within two standard deviations that the mean. Approximately 99.7% that the data values loss within 3 standard deviations that the mean.The empirical dominance will it is in revisited later on in the thing on regular probabilities.

## Using the TI-82 to find these values

You may use the TI-82 to uncover the actions of central tendency and the actions of variationusing the list managing capabilities of the calculator.Table of Contents