Standard Deviation Using a Calculator

Let the Calculator Handle the Repeated Arithmetic

In Calculating Standard Deviation by Hand, you found the sample standard deviation by calculating deviations from the mean, squaring them, and using the \(n-1\) denominator. A calculator can carry out those steps quickly for a longer list of observations. You still need to enter the data correctly, identify the right result, and explain what it describes.

On a TI-84, the one-variable statistics function is called 1-Var Stats. Its output includes both \(Sx\) and \(\sigma x\). These are not two different ways to label the same result: \(Sx\) is the sample standard deviation, while \(\sigma x\) is the population standard deviation. The calculator displays both, so you must choose based on what your data represent.

Definition: \(Sx\) is the sample standard deviation, calculated using \(n-1\) in the denominator. \(\sigma x\) is the population standard deviation, calculated using \(N\) in the denominator when the observations describe the entire population of interest.

Here, \(n\) is the number of observations in a sample, and \(N\) is the number of values in the entire population being described. The symbol \(\sigma\) is also used for a population standard deviation in statistics; on the calculator screen, the label is typically \(\sigma x\). A calculator cannot decide whether your list is a sample or a full population. That choice depends on the question and the group you mean to describe.

For sample data, \(Sx\) corresponds to the formula from the hand-calculation tutorial. The calculator’s \(\sigma x\) uses the sum of squared deviations divided by the number of observations instead. Because \(n-1\) is smaller than \(n\), \(Sx\) will be larger than \(\sigma x\) for data with some spread. If all observations are identical, both values are zero.

Sx=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}} \qquad \sigma x=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}

Entering Data and Reading 1-Var Stats

For a basic TI-84 calculation, enter each observation into a list, such as L1. Press STAT, choose EDIT, and type the data values into the L1 column, pressing ENTER after each value. Then press STAT, move to CALC, choose 1-Var Stats, select L1, and press ENTER. Menu wording and key presses may vary slightly across calculator models, but the goal is the same: select the list containing the observations and run the one-variable statistics summary.

Check that the list contains every observation exactly once. A repeated value should appear repeatedly if it occurred repeatedly in the data. Selecting a list that is empty, incomplete, or left over from another problem can produce an answer that looks plausible but does not describe the intended data.

A typical TI-84 output includes several statistics. For this tutorial, the most useful labels are:

Calculator label	What it reports
\(\bar{x}\)	The mean of the values in the selected list
\(\Sigma x\)	The sum of the values in the selected list
\(\Sigma x^2\)	The sum of the squared values in the selected list
\(Sx\)	The sample standard deviation
\(\sigma x\)	The population standard deviation
\(n\)	The number of values in the selected list

The symbols \(\Sigma x\) and \(\Sigma x^2\) are useful checks, but they are not themselves standard deviations. In particular, \(\Sigma x^2\) means “add the squared observations”; it is not the sum of squared deviations from the mean. When the question asks for the sample standard deviation, find \(Sx\), not \(\sigma x\).

Worked Examples: Choosing the Right Calculator Result

Worked Example: Sample of Seedling Heights

A fictional sample of five seedlings has heights of 8, 11, 13, 14, and 19 centimeters. Use 1-Var Stats to find the sample standard deviation, and identify the population standard deviation the calculator also reports.

Enter and summarize the observations. Put 8, 11, 13, 14, and 19 in L1. Run 1-Var Stats for L1. The output gives \(n=5\), \(\bar{x}=13\), \(Sx\approx4.0620\), and \(\sigma x\approx3.6332\). The sample is the group of interest in the question, so report \(Sx\).

Check the calculator’s results. The mean is \((8+11+13+14+19)/5=65/5=13\) centimeters. The deviations from 13 are \(-5,-2,0,1,6\); their squares sum to \(25+4+0+1+36=66\) square centimeters. Therefore, the sample and population versions are:

Sx=\sqrt{\frac{66}{5-1}} =\sqrt{16.5} \approx4.0620\text{ centimeters}

\sigma x=\sqrt{\frac{66}{5}} =\sqrt{13.2} \approx3.6332\text{ centimeters}

The sample standard deviation of the seedlings’ heights is about 4.0620 centimeters. The calculator’s \(\sigma x\) is smaller because it divides the same sum of squared deviations by 5 rather than 4. Since the data are being treated as a sample, \(Sx\) is the result that answers the question.

Worked Example: Every Shift in a Small Event

A fictional community event has exactly five scheduled information-desk shifts. The number of calls handled during those shifts was 3, 5, 5, 7, and 10. Find the population standard deviation for those five shifts using 1-Var Stats.

State. The group of interest is all five shifts at this event, not a sample intended to represent additional shifts. The population standard deviation \(\sigma x\) is appropriate.

Plan and check the data. Enter 3, 5, 5, 7, and 10 in L1 and run 1-Var Stats. The repeated 5 belongs in the list twice because two shifts each handled five calls. The output should show \(n=5\) and \(\bar{x}=6\). It also reports \(Sx\approx2.6458\) and \(\sigma x\approx2.3664\).

Do. The values add to 30, so the mean is \(30/5=6\) calls per shift. The deviations from 6 are \(-3,-1,-1,1,4\), and their squares add to \(9+1+1+1+16=28\). Because the five observed shifts are the entire population being described, divide by 5:

\sigma x=\sqrt{\frac{28}{5}} =\sqrt{5.6} \approx2.3664\text{ calls per shift}

For comparison, the sample standard deviation shown on the same screen is \(\sqrt{28/(5-1)}=\sqrt{7}\approx2.6458\) calls per shift. It is a valid statistic for these data, but it is not the requested population standard deviation. The population standard deviation in the number of calls handled across all five scheduled shifts is about 2.3664 calls per shift.

Worked Example: The Same List, a Different Question

A fictional student records the number of minutes spent on a technology task on four randomly selected evenings: 2, 4, 6, and 9 minutes. Use 1-Var Stats to find the standard deviation that describes the sample, then explain which result would be used to describe only those four recorded evenings.

Run the calculator summary. Enter 2, 4, 6, and 9 in L1 and run 1-Var Stats. The screen reports \(n=4\), \(\bar{x}=5.25\), \(Sx\approx2.9861\), and \(\sigma x\approx2.5860\). The four evenings were selected from a broader set of evenings, so use \(Sx\) when describing the sample.

Check the arithmetic. The sum is \(2+4+6+9=21\), and \(21/4=5.25\) minutes. The deviations are \(-3.25,-1.25,0.75,\) and \(3.75\) minutes. Their squares sum to \(10.5625+1.5625+0.5625+14.0625=26.75\) minutes squared. Thus:

Sx=\sqrt{\frac{26.75}{4-1}} =\sqrt{8.9167} \approx2.9861\text{ minutes}

If the question instead asks for the population standard deviation of just the four recorded evenings, use the denominator 4:

\sigma x=\sqrt{\frac{26.75}{4}} =\sqrt{6.6875} \approx2.5860\text{ minutes}

The sample standard deviation is about 2.9861 minutes for the four sampled evenings. The population standard deviation of those four recorded values is about 2.5860 minutes. The list itself did not change; what changed was the group the question asks you to describe.

How to Decide Between \(Sx\) and \(\sigma x\)

The decision comes from the role of the data, not from which number looks more useful or which one appears first on the calculator screen. If the observations are a sample, use \(Sx\) to report the sample standard deviation. If the observations include every member of the specific population being described, use \(\sigma x\).

Be precise about the population. A complete list of all observations in one small event can be a population for a question about that event, even if the event itself is only one of many possible events. If the goal is to use observations from that event to describe a broader group of events, those observations are a sample for that broader question. The intended group determines which version fits.

A quick check is to compare the result with the denominator. For \(n\) observations, \(Sx\) uses \(n-1\); \(\sigma x\) uses \(n\). Do not confuse the population formula with the sample formula from Calculating Standard Deviation by Hand. The calculator reports both, but it does not automatically know which one a question requires.

Common Mistakes and AP Exam Tips

Reporting \(\sigma x\) when the question asks for a sample standard deviation. Read the question’s group carefully. If the data are a sample, identify \(Sx\) and state it in context.
Assuming the calculator knows whether the data are a sample. It does not. It calculates both versions from the list, so the student must choose based on the group being described.
Entering a value incorrectly or omitting a repeat. Check the data list against the original observations. A missing or extra value changes the sample size, mean, and standard deviation.
Misreading \(\Sigma x^2\) as the sum of squared deviations. It is the sum of the observations squared. To check \(Sx\) by hand, calculate deviations from \(\bar{x}\), square them, and add those squares.
Reporting a number without units or context. Standard deviation has the same units as the original measurements. A full-credit statement identifies the group, variable, value, and units.
Rounding intermediate work too soon. Use the calculator’s unrounded computation and round the final result to a sensible number of decimal places for the situation.

A complete calculator response should name the statistic chosen and explain why it fits. For example: “The sample standard deviation of the five sampled seedling heights is about 4.062 centimeters, so I use \(Sx\).” If the data are the entire population in question, state that and report \(\sigma x\) instead.

Key takeaway: Enter each observation correctly and run 1-Var Stats. Use \(Sx\) for a sample standard deviation and \(\sigma x\) for the standard deviation of the entire population being described. The calculator provides both; the question’s group tells you which to report.

Check Your Understanding

Use the calculator labels and the group described in each question to choose the appropriate result.

A TI-84 output shows \(Sx=6.2\) and \(\sigma x=5.8\). Which value is the sample standard deviation?
A list contains 6, 8, and 10. Find the mean and the sum of squared deviations, then calculate both \(Sx\) and \(\sigma x\).
A researcher records every one of the 12 machines in a particular workshop. For a question about those 12 machines, which calculator result is appropriate: \(Sx\) or \(\sigma x\)? Explain.
A student enters five measurements but accidentally omits a repeated value. Name two parts of the 1-Var Stats output that could change.
Explain why a calculator can display both \(Sx\) and \(\sigma x\) without knowing which value answers a particular question.