2.8 Measuring the spread of the data (modified r. bloom)  (Page 4/6)

The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.

Notice that instead of dividing by n=20, the calculation divided by n-1=20-1=19 because the data is a sample. For the sample variance, we divide by the sample size minus one ( $\mathrm{n-1}$ ). Why not divide by $n$ ? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by $\mathrm{(n-1)}$ gives a better estimate of the population variance.

The standard deviation, $s$ or $\sigma$ , is either zero or larger than zero. When the standard deviation is 0, there is no spread; that is, the all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make $s$ or $\sigma$ very large.

The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better "feel" for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data .

Comparing values from different data sets

• For each data value, calculate how many standard deviations the value is away from its mean.
• Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.
• $\#\mathrm{ofSTDEVs}=\frac{\mathrm{value}-\mathrm{mean}}{\mathrm{standard\; deviation}}$
• Compare the results of this calculation.

#ofSTDEVs is often called a "z-score"; we can use the symbol z. In symbols, the formulas become:

The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.

For any data set, no matter what shape the distribution of the data is:

• At least 75% of the data is within 2 standard deviations of the mean.
• At least 89% of the data is within 3 standard deviations of the mean.
• At least 95% of the data is within 4 1/2 standard deviations of the mean.
• This is known as Chebyshev's Rule.

For data having a distribution that is mound-shaped and symmetric:

• Approximately 68% of the data is within 1 standard deviation of the mean.
• Approximately 95% of the data is within 2 standard deviations of the mean.
• More than 99% of the data is within 3 standard deviation of the mean.
• This is known as the Empirical Rule.
• It is important to note that this rule only applies when the shape of the distribution of the data is mound-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.