Using 1-Var Stats for a Probability Distribution
In Interpreting Standard Deviation of a Random Variable, you learned to describe \(\sigma_X\) as the typical distance of the values of \(X\) from their mean. In the earlier tutorials on the mean, variance, and standard deviation of a discrete random variable, you calculated those summaries from a probability table. Here, you will use a calculator’s 1-Var Stats command to check those hand calculations.
The key is to use the probability list as a frequency list. Enter the possible values of \(X\) in one list and their probabilities in a second list. When you tell the calculator to use the second list as frequencies, each value is weighted by its probability. The calculator’s population mean and population standard deviation then match \(\mu_X\) and \(\sigma_X\).
The order matters: the probability in each row must stay beside its matching value. Before calculating, check that each probability is between 0 and 1 and that the probabilities add to 1, as in Checking Whether a Probability Distribution Is Valid.
Entering the Lists and Reading the Output
On a TI-84, open the list editor with STAT, then 1:Edit. Enter the possible values in L1 and their probabilities in L2, keeping the rows aligned. To run the calculation, choose STAT, move to CALC, and select 1:1-Var Stats. Specify both lists, in the form 1-Var Stats L1,L2, and press ENTER. Menu names can vary slightly by calculator model, but the essential instruction is to include L2 as the frequency list.
The calculator reports several quantities. For a probability distribution, the most important are \(\bar{x}\) and \(\sigma_x\). The calculator’s \(\bar{x}\) is the weighted mean, and \(\sigma_x\) is the population standard deviation. The output \(S_x\), by contrast, is a sample standard deviation. It is not the standard deviation of the probability distribution.
You may also see \(n\), \(\Sigma x\), and \(\Sigma x^2\). With a frequency list, \(n\) is the total of the frequency weights, not the number of rows. For probability weights that add to 1, \(n=1\). Likewise, \(\Sigma x\) and \(\Sigma x^2\) are weighted totals: they correspond to \(\sum xp\) and \(\sum x^2p\), not to adding the values in L1 without weights.
Because a probability list sums to 1, \(S_x\), which uses a sample denominator based on \(n-1\), may be undefined or unhelpful on a calculator. Do not use \(S_x\) for this task. If the calculator does not display \(\sigma_x\) when the probabilities are entered directly, multiply every probability by the same positive number, such as 100, and use those scaled values as frequency weights. The population standard deviation is unchanged because the same multiplier appears in both the numerator and denominator of its weighted calculation. The scaled entries are weights, not probabilities, so use \(\sigma_x\), not \(S_x\), and do not interpret the new \(n\) as a probability total.
Worked Example: Messages Received by a Help Desk
Worked Example: Messages Received by a Help Desk
Let \(X\) be the number of messages a help desk receives during a randomly selected 10-minute period. The model assigns probabilities 0.20, 0.50, and 0.30 to 0, 1, and 2 messages, respectively. Find the mean and standard deviation by hand, then describe how 1-Var Stats checks the results.
| Messages \(x\) | \(P(X=x)\) |
|---|---|
| 0 | 0.20 |
| 1 | 0.50 |
| 2 | 0.30 |
Check the distribution and find the mean. Each probability is between 0 and 1, and \(0.20+0.50+0.30=1.00\). Using the probability-weighted mean from The Mean of a Discrete Random Variable:
Find the standard deviation by hand. Use \(\mu_X=1.10\) to calculate each weighted squared deviation, as in The Standard Deviation of a Discrete Random Variable.
| \(x\) | \(P(X=x)\) | \(x-\mu_X\) | \((x-\mu_X)^2\) | \((x-\mu_X)^2P(X=x)\) |
|---|---|---|---|---|
| 0 | 0.20 | \(-1.10\) | 1.21 | 0.242 |
| 1 | 0.50 | \(-0.10\) | 0.01 | 0.005 |
| 2 | 0.30 | 0.90 | 0.81 | 0.243 |
The weighted squared deviations sum to \(0.242+0.005+0.243=0.490\). Taking the nonnegative square root gives:
Check with 1-Var Stats. Enter 0, 1, and 2 in L1, and enter 0.20, 0.50, and 0.30 in the corresponding rows of L2. Run 1-Var Stats with L1,L2. The output should show \(\bar{x}=1.10\) and \(\sigma_x=0.700\), matching the hand calculations. The weighted totals provide additional checks: \(\Sigma x=0(0.20)+1(0.50)+2(0.30)=1.10\), and \(\Sigma x^2=0^2(0.20)+1^2(0.50)+2^2(0.30)=1.70\).
Interpret. According to this model, the number of messages in a 10-minute period typically differs from its mean of 1.10 messages by about 0.70 message.
Worked Example: Time to Complete a Setup
Worked Example: Time to Complete a Setup
A technician models \(T\), the time in minutes to complete a particular setup. The possible times are 2, 5, and 9 minutes, with probabilities 0.25, 0.50, and 0.25. Find \(\mu_T\) and \(\sigma_T\), and then use 1-Var Stats to check both.
Check the probabilities and calculate the mean. The probabilities are all between 0 and 1 and sum to \(0.25+0.50+0.25=1.00\).
Calculate the weighted squared deviations.
| \(t\) | \(P(T=t)\) | \(t-\mu_T\) | \((t-\mu_T)^2\) | \((t-\mu_T)^2P(T=t)\) |
|---|---|---|---|---|
| 2 | 0.25 | \(-3.25\) | 10.5625 | 2.640625 |
| 5 | 0.50 | \(-0.25\) | 0.0625 | 0.031250 |
| 9 | 0.25 | 3.75 | 14.0625 | 3.515625 |
Add the final column: \(2.640625+0.031250+3.515625=6.187500\) minutes squared. Therefore:
Check with 1-Var Stats. Put 2, 5, and 9 in L1, and 0.25, 0.50, and 0.25 in L2. The calculator should report \(\bar{x}=5.25\) and \(\sigma_x\approx2.4875\). As another check, the weighted squared total is \(\Sigma t^2=2^2(0.25)+5^2(0.50)+9^2(0.25)=1+12.5+20.25=33.75\). Subtracting the square of the mean gives \(33.75-(5.25)^2=33.75-27.5625=6.1875\), the same variance found from the weighted deviations.
The output \(S_x\) is not the value to report. The distribution’s standard deviation is \(\sigma_T\), about 2.4875 minutes.
Worked Example: Equipment Checks per Shift
Worked Example: Equipment Checks per Shift
A model for \(T\), the number of equipment checks completed during a shift, gives probabilities 0.20, 0.50, and 0.30 for 4, 7, and 10 checks. Find the mean and standard deviation by hand, then specify the calculator entries and the output to use.
| Checks \(t\) | \(P(T=t)\) |
|---|---|
| 4 | 0.20 |
| 7 | 0.50 |
| 10 | 0.30 |
Find the mean. The probabilities are between 0 and 1 and total 1.00.
Find the variance and standard deviation.
| \(t\) | \(P(T=t)\) | \(t-\mu_T\) | \((t-\mu_T)^2\) | \((t-\mu_T)^2P(T=t)\) |
|---|---|---|---|---|
| 4 | 0.20 | \(-3.30\) | 10.89 | 2.178 |
| 7 | 0.50 | \(-0.30\) | 0.09 | 0.045 |
| 10 | 0.30 | 2.70 | 7.29 | 2.187 |
The weighted squared deviations sum to \(2.178+0.045+2.187=4.410\). Thus:
Check with 1-Var Stats. Enter 4, 7, and 10 in L1 and 0.20, 0.50, and 0.30 in L2, then run 1-Var Stats L1,L2. The calculator’s \(\bar{x}\) should be 7.30 and its \(\sigma_x\) should be 2.100. The weighted square total confirms the variance: \(\Sigma t^2=4^2(0.20)+7^2(0.50)+10^2(0.30)=3.20+24.50+30.00=57.70\), so \(57.70-(7.30)^2=57.70-53.29=4.41\). The square root of 4.41 is 2.10.
In context, the number of checks completed in a shift typically differs from the mean of 7.30 checks by about 2.10 checks.
Common Mistakes and AP Exam Tips
- Leaving out the frequency list. Running 1-Var Stats with only L1 treats each listed value as having equal weight. Include L2 so the calculation uses the distribution’s probabilities.
- Misaligning a value and its probability. The first probability must match the first value, the second probability the second value, and so on. Check the rows before running the command.
- Reporting \(S_x\) instead of \(\sigma_x\). \(S_x\) is the calculator’s sample standard deviation. For the probability distribution, report \(\sigma_x\), which corresponds to \(\sigma_X\).
- Reading \(n\) as the number of possible values. When using probabilities as weights, \(n\) is their total, usually 1. It is not the number of rows in the distribution table.
- Treating \(\Sigma x^2\) as an unweighted sum. With a frequency list, the calculator’s \(\Sigma x^2\) is the weighted total \(\sum x^2P(X=x)\). Check it by multiplying each squared value by its probability.
- Rounding too early. Keep enough digits in the hand calculation, especially before squaring deviations or taking a square root. Round the final result consistently with the requested precision.
A full-credit explanation identifies the correct calculator statistic and connects it to the random variable. For example: “The calculator’s \(\sigma_x\) is 2.10, so the standard deviation of the modeled number of checks per shift is 2.10 checks.” If asked to show work, include the weighted mean and the weighted squared deviations; a calculator result alone may not communicate how the answer was obtained.
Key Takeaway
1-Var Stats can check the hand-calculated mean and standard deviation of a discrete probability distribution when the values and their probabilities are entered as a data list and a frequency list. Match each value with its probability, use \(\bar{x}\) for the mean and \(\sigma_x\) for the distribution standard deviation, and verify that the probabilities form a valid distribution.
Check Your Understanding
Let \(Y\) be the number of small repairs needed during a randomly selected service period. Use the model below.
| Repairs \(y\) | \(P(Y=y)\) |
|---|---|
| 0 | 0.10 |
| 2 | 0.60 |
| 5 | 0.30 |
- Check that the table gives a valid probability distribution.
- Calculate \(\mu_Y\) by hand, showing each value-times-probability product.
- Calculate \(\sigma_Y^2\) and \(\sigma_Y\) by hand, showing the weighted squared deviations.
- State what belongs in L1 and L2, and write the 1-Var Stats command using both lists.
- Which calculator output should be used for the distribution standard deviation, \(\sigma_x\) or \(S_x\)? Explain briefly.