### Learning Outcomes

Recognize, describe, and calculate the measures of the spread of data: variance, standard deviation, and range.You are watching: The sum of the deviations from the mean will always be equal to ______________.

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The **standard deviation** is a number that measures how far data values are from their mean.

The standard deviation provides a numerical measure of the overall amount of variation in a data set, and can be used to determine whether a particular data value is close to or far from the mean.

The standard deviation provides a measure of the overall variation in a data set.The standard deviation is always positive or zero. The standard deviation is small when the data are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.

Suppose that we are studying the amount of time customers wait in line at the checkout at supermarket

Because supermarket

Suppose that Rosa and Binh both shop at supermarket

Rosa waits for seven minutes:

Seven is two minutes longer than the average of five; two minutes is equal to one standard deviation.Rosa’s wait time of seven minutes is**two minutes longer than the average**of five minutes.Rosa’s wait time of seven minutes is

**one standard deviation above the average**of five minutes.

Binh waits for one minute.

One is four minutes less than the average of five; four minutes is equal to two standard deviations.Binh’s wait time of one minute is**four minutes less than the average**of five minutes.Binh’s wait time of one minute is

**two standard deviations below the average**of five minutes.

A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if it is more than two standard deviations away is more of an approximate “rule of thumb” than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is further away than two standard deviations. (You will learn more about this in later chapters.)

The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is**one** standard deviation to the **right** of five because

If one were also part of the data set, then one is **two** standard deviations to the **left** of five because

In general, a

**va**

**lue = mean + (#ofSTDEV)(standard deviation)**where #ofSTDEVs = the number of standard deviations#ofSTDEV does not need to be an integerOne is

**two s**

**tandard deviations less than the mean**of five because:

The equation **value = mean + (#ofSTDEVs)(standard deviation)** can be expressed for a sample and for a population.

The lower case letter

The symbol

## Calculating the Standard Deviation

If **deviation**. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is

The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter

To calculate the standard deviation, we need to calculate the variance first. The**variance** is the **average of the squares of the deviations** (the

If the numbers come from a census of the entire **population** and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by **sample** rather than a population, when we calculate the average of the squared deviations, we divide by ** n – 1**, one less than the number of items in the sample.

In the following video an example of calculating the variance and standard deviation of a set of data is presented.

## Formulas for the Sample Standard Deviation

For the sample standard deviation, the denominator is

## Formulas for the Population Standard Deviation

For the population standard deviation, the denominator is

In these formulas,

## Sampling Variability of a Statistic

How much the statistic varies from one sample to another is known as the **sampling variability of a statistic**. You typically measure the sampling variability of a statistic by its standard error. The **standard error of the mean** is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean when you learn about The Central Limit Theorem (not now). The notation for the standard error of the mean is

### Note

**In practice, use a calculator or computer software to calculate the standard deviation. If you are using a TI-83, 83+, 84+ calculator, you need to select the appropriate standard deviation σ_x or s_x from the summary statistics.** We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. (The calculator instructions appear at the end of this example.)

### Example

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a sample ofSolution:

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating

( | ( | ( | ||

The total is |

The sample variance,

The **sample standard deviation**

**Typically, you do the calculation for the standard deviation on your calculator or computer.** The intermediate results are not rounded. This is done for accuracy.

**value = mean + (#ofSTDEVs)(standard deviation)**. Verify the mean and standard deviation or a calculator or computer. For a sample:

**from**(below and above) the mean.

USING THE TI-83, 83+, 84, 84+ CALCULATORClear lists L1 and L2. Press STAT 4:ClrList. Enter 2nd 1 for L1, the comma (,), and 2nd 2 for L2.Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down.Put the data values (

## Explanation of the standard deviation calculation shown in the table

The deviations show how spread out the data are about the mean. The data value **If you add the deviations, the sum is always zero. **(For Example 1, there are

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 **sample** variance, we divide by the sample size minus one (**The sample variance is an estimate of the population variance.** Based on the theoretical mathematics that lies behind these calculations, dividing by (

### Note

Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number which measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic.

The standard deviation,

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**. Display your data in a histogram or a box plot.

### Example

Use the following data (first exam scores) from Susan Dean’s spring pre-calculus class:

DataFrequencyRelative FrequencyCumulative Relative Frequency

The long left whisker in the box plot is reflected in the left side of the histogram. The spread of the exam scores in the lower

### Try It

The following data show the different types of pet food stores in the area carry.

## Standard Deviation of Grouped Frequency Tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula:

*Mean of Frequency Table* =

where

Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how “unusual” individual data is compared to the mean.

### Example

Find the standard deviation for the data in the table below.

ClassFrequency,For this data set, we have the mean,

### Try It

Find the standard deviation for the data from the previous example

ClassFrequency,First, press the **STAT **key and select **1:Edit**

Input the midpoint values into **L1** and the frequencies into **L2**

## Comparing Values from Different Data Sets

The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.

For each data value, calculate how many standard deviations away from its mean the value is.Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.*#ofSTDEVs*=

#ofSTDEVs is often called a ”

Sample | ||

Population |

### Example

Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?

StudentGPASchool Mean GPASchool Standard DeviationJohn | |||

Ali |

For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.

For John,

For Ali,

John has the better GPA when compared to his school because his GPA is **below** his school’s mean while Ali’s GPA is **below** his school’s mean.

John’s

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 the distribution of the data is:**

**For data having a distribution that is BELL-SHAPED and SYMMETRIC:**

## Concept Review

The standard deviation can help you calculate the spread of data. There are different equations to use if are calculating the standard deviation of a sample or of a population.

The Standard Deviation allows us to compare individual data or classes to the data set mean numerically.*μ*, and the formula

## Formula Review

where

## References

Data from Microsoft Bookshelf.

See more: How To Breed All Gemstone Dragons In Dragonvale ? : Dragonvale

King, Bill.”Graphically Speaking.” Institutional Research, Lake Tahoe Community College. Available online at http://www.ltcc.edu/web/about/institutional-research (accessed April 3, 2013).