Mean, SD, t-test

E1: D1 paired instructional dataset

D1 is an invented numerical practice dataset. It is not the missing March 4 raw lab table. Do not collect pulse readings or perform an intervention for this task.

The modeled variable is pulse rate in beats per minute, abbreviated bpm. A and B are labels for two conditions; no treatment or physical procedure is assigned.

Each row represents a different fictional participant with one value in A and one in B. The two values in a row are paired. Each condition has five values; there are five pairs, not ten independent participants.

P1: A = 66 bpm; B = 68 bpm.

P2: A = 68 bpm; B = 70 bpm.

P3: A = 70 bpm; B = 74 bpm.

P4: A = 72 bpm; B = 76 bpm.

P5: A = 74 bpm; B = 82 bpm.

For the modeled design, participants are distinct and no participant is repeated across rows. Treat pairs as independent for the structure exercise. Within a pair, A and B are linked.

No random sampling, randomized condition order or distribution for the population of paired differences is supplied. The values do not establish those assumptions. No real population was sampled.

E2: Mean and sample SD formula sheet

Use the sample SD convention today. Mean = sum of values divided by n. Here n is the number of values within the condition.

Sample SD = square root of [sum of (each value minus mean) squared, divided by (n minus 1)]. This formula needs at least two values.

In symbols written as text: x is a value, x-bar is the sample mean, n is the count, and s is sample SD. Sum means add every indicated term.

Step 1: add the values and divide by n to find the mean. Step 2: subtract that mean from each value to find each deviation.

Step 3: square each deviation by multiplying it by itself. Step 4: add the squared deviations.

Step 5: divide that sum by n minus 1. This is sample variance. Step 6: take its square root to obtain sample SD.

A square root reverses squaring: the square root of 4 is 2. Variance has squared units; SD returns to the original units.

Population SD uses n in the divisor when describing an entire defined population. Do not switch formulas mid-calculation. This practice requests sample SD even though its numbers are invented.

Keep exact values or several decimal places during calculation. Round final mean and SD to two decimals and include units. A calculator is allowed; retain the steps.

E3: What a t-test can and cannot answer

A t-test assesses evidence about a population mean or mean difference under a stated hypothesis and assumptions. It does not label a difference real or imaginary.

For linked A/B observations, a paired t-test uses each within-pair difference. It asks about the population mean difference, often compared with zero.

Independent two-sample t-tests concern separate groups. Their uncertainty depends on both group sizes and spread. They must not treat repeated readings from one person as independent people.

For a paired t-test with a small sample, independent pairs and an approximately normal population of differences are important assumptions. Normal here means a bell-shaped population distribution.

D1 has only five pairs. Its design points toward a paired approach, but no population-difference distribution or sampling basis is supplied. Do not claim a justified population test or significant result from D1.

To decide, check whether each A value is linked to one B value in the same row. A linked design calls for comparing the within-row differences, not two independent groups. Then name each assumption D1 does not supply.

No t statistic or probability calculation is required today. Explain purpose, design fit, n and what is still unknown. Describing the dataset does not require running an inferential test.

Warmup and recovery inputs

W1 warmup: 2, 4, 6, 8, 10 count-units. These five invented values are separate from D1. Find their mean by hand, then describe their spread before calculating any SD.

Spread needs a comparison or scale. Name the distance between low and high values rather than calling spread large without context. SD describes spread around the mean; it does not by itself show measurement quality.

R1 recovery: 8, 10, 12 count-units, with the mean given as 10 and n = 3. Use E2 to compute sample SD and explain what a larger SD would mean.

R1 is a short recovery practice. Return to both D1 conditions for the full product unless your teacher explicitly assigns a different scope.

Work independently with the same supplied numbers and a calculator or spreadsheet. If using a spreadsheet, show the values, mean, deviations, squared deviations, sum and divisor; a single output cell is not all the steps.

Checks and optional comparison

Check C1 is a separate hypothetical summary: A mean = 81, SD = 3; B mean = 75, SD = 3. No units, sample sizes, pairing or sampling design are provided.

The mean gap is 6, twice the stated SD. Before judging the claim, list what a significance, overlap, importance or cause judgment would need, and check whether C1 supplies it.

Optional extension X1: compare invented independent groups X = 9, 10, 11 and Y = 6, 10, 14 count-units. Each has n = 3 and mean = 10; their sample SDs are 1 and 4.

For this mathematical comparison only, assume independent observations from normal populations. The two group means are equal, so their observed mean gap is zero. Different spread does not turn that zero gap into evidence of a mean difference.

Explain why the original question “which spread makes this a real difference?” is not valid for equal means. If a nonzero gap and n were held fixed, spread would affect uncertainty. That is a different comparison.

No actual population effect is established by X1. Do not run a full test for this optional explanation.

Keep original practice and prior work separate

Use D1 for this original statistics practice. If your teacher instead requires your actual authorized lab table, ask for that table and its participant/trial design before analyzing it.

Do not label D1 as data you collected. It does not finish the March 4 sensor lab or establish that its raw table is complete.

The weekly Research design ticket and Study critique packet remain separate work. Check their actual materials, instructions and whether Schoology shows Submitted with your teacher. This calculation record does not complete them.

The official statistics activity and Activity 2.1.2 directions also need teacher confirmation. Save this practice separately; do not mark official work complete merely because these calculations are finished.

W1 mean and spread prediction

Find the mean of the five W1 values by hand. Describe their spread using the low and high values.

Show the sum divided by five. This warmup is separate from D1.

D1 means, A and B

For each condition, list the five values, add them, state n and divide to find the mean. Include bpm.

Keep A and B separate. Each condition has five values.

D1 A sample SD steps

Show each A value minus its mean, each squared deviation, their sum, the sample divisor, variance and sample SD.

Use the n-minus-1 formula. Preserve units and round only the final SD to two decimals.

D1 B sample SD steps

Repeat the full sample SD sequence for B. Compare its spread with A using the SD values.

Use B’s own mean. A larger SD describes greater spread here, not proof that the measurements are wrong.

t-test purpose and applicability

Explain what a t-test compares. For D1, name the relevant paired design, its n, and why the missing assumptions block a population conclusion.

Paired differences, not ten independent people. No t statistic or significance calculation is required.

C1 check and SD exit statement

Explain why C1’s six-point gap and SDs alone cannot settle importance or significance. Then state what a larger SD tells you about values in the same units.

List what C1 does not report that such a judgment would need. Keep both answers in the same calculation record.