data-analysis | High School Statistics Textbook

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Appendix 5 — Technology Tips (On Your Phone & Laptop)

Statistics can be done with calculators, spreadsheets, or software. Here’s a quick guide.

Excel / Google Sheets

Task	Formula	Example
Mean	`=AVERAGE(A1:A10)`	Mean of scores in A1–A10
Standard Deviation	`=STDEV.S(A1:A10)`	Spread of scores
t-test	`=T.TEST(A1:A10,B1:B10,2,2)`	Compare two groups

R (RStudio or RStudio Cloud)

Task	Command	Example
Mean	`mean(x)`	`mean(c(6,8,10)) = 8`
SD	`sd(x)`	`sd(c(6,8,10)) = 2`
t-test	`t.test(x,y)`	Compare two groups

Python (NumPy / SciPy / Pandas)

Task	Command	Example
Mean	`np.mean(x)`	`np.mean([6,8,10]) = 8`
SD	`np.std(x, ddof=1)`	`np.std([6,8,10],ddof=1) = 2`
t-test	`stats.ttest_ind(x,y)`	Compare two groups

iPhone Calculator

Rotate sideways → scientific mode
Use √ for square root
Parentheses matter: type numerator, then divide by denominator
Fine for small problems, but not for full datasets

Summary

For quick homework: iPhone calculator
For assignments: Excel / Google Sheets
For coding: Python (Colab) or R (RStudio Cloud)

📱 QR: Open sample data in Google Sheets (ready to practice mean, SD, t-test)

Visuals

Figure E.1 — Screenshots of the same mean calculation in Sheets, R, and Python side by side.

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.

Applications: Cases and Examples

Case 1 — Independent t-test (Two Groups)

Scenario: A teacher wants to compare math test scores between students taught with traditional lectures and those taught with interactive software.

Question: Are the two teaching methods different in average test score?

Design/Test: Independent-samples t-test.

Worked Example:

Group A (Lecture): mean = 78, SD = 10, n = 20
Group B (Software): mean = 85, SD = 12, n = 20

Formula:
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$

In words:
$$t = \frac{\text{mean}_1 - \text{mean}_2}{\sqrt{\tfrac{\text{variance}_1}{n_1} + \tfrac{\text{variance}_2}{n_2}}}$$

Plugging in values:
$$t = \frac{78 - 85}{\sqrt{\tfrac{100}{20} + \tfrac{144}{20}}} = \frac{-7}{\sqrt{5 + 7.2}} = \frac{-7}{\sqrt{12.2}} = \frac{-7}{3.49} = -2.01$$

Degrees of freedom = 38.

Case 2 — Paired t-test (Before and After)

Scenario: Students take a memory test before and after a week of practice.

Question: Did memory scores improve after training?

Design/Test: Paired-samples t-test.

Worked Example:

Differences (After – Before): 2, 4, 3, 5, 6

Mean difference:
$$\bar{D} = \frac{2+4+3+5+6}{5} = 4$$
Standard deviation of differences: $$s_D = 1.58$$

Formula:
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$

Plugging in values:
$$t = \frac{4}{1.58/\sqrt{5}} = \frac{4}{0.71} = 5.63$$

Degrees of freedom = 4.

Case 3 — One-way ANOVA (Three Groups)

Scenario: A psychologist tests three methods of stress reduction: meditation, exercise, and music.

Question: Do the methods differ in average stress score?

Design/Test: One-way ANOVA.

Worked Example (summary):

Group means: Meditation = 65, Exercise = 70, Music = 80
$$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
$$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$

Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

Plugging in values:
$$F = \frac{150}{16.7} = 9.0$$

df = (2, 12).

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.