[Note: Version A of the Monitoring Your Readiness page-part is the full version, containing an overview and every learning objective found in both pages of the section. Version B is a less detailed version that just highlights the most important objectives required for the self-check.
Also, the tables don’t appear efficiently on the page and hopefully can be made to appear less huge and overwhelming.]

(A) Monitoring Your Readiness 

In this section’s What to Know page, you explored variability (spread) of a dataset by visually assessing histograms and dotplots. You also learned how to calculate standard deviation, variance, and range, and to compare the variability of multiple distributions. In the Forming Connections activity, you learned to use technology to describe the variability of data and explored the effect of outliers on the standard deviation data.

The following table lists the learning goals for this section. Reflect for a moment upon how you answered the questions posed in the activities. Then, for each of the objectives listed in the table, indicate your level of comfort being able to perform them on a formal assessment. Use the following descriptions to rate yourself:

5 — I am extremely confident I can do this task.

4 — I am somewhat confident I can do this task.

3 — I am not sure how confident I am.

2 — I am not very confident I can do this task.

1 — I am definitely not confident I can do this task.

Learning Goals: I can …	Questions to check your understanding	Rating: from 1 (not confident) to 5 (extremely confident)
Compare the variability of multiple datasets visually using histograms	What to Know Question 1
Compare the variability of multiple datasets visually using dotplots.	What to Know Question 2, 3
Use a data analysis tool to identify the standard deviation of a dataset.	What to Know Question 4 – 6*
Calculate the variance of a dataset given standard deviation	What to Know Question 7
Use a data analysis tool to calculate variability by identifying the range of a dataset.	What to Know Question 8
Use a data analysis tool to describe variability of data.	Forming Connections Questions 1 – 4
Find and interpret the standard deviation of data.	Forming Connections Questions 5 – 7
*Note that in What to Know Questions 4 and 5, you also refreshed your understanding of population and sample, mean and median.

The following summary of this section’s concepts and skills may be helpful to boost your confidence before trying the self-check.

(B) Monitoring Your Readiness

Before you attempt the self-check, think about what you know and what is still challenging. The following statements identify key skills to take away from the material in this section. For each of them, rate how confident you are that you can successfully do that skill. Use the following descriptions to rate yourself:

5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.

Learning Goals: I can …	Rating: from 1 (not confident) to 5 (extremely confident)
Visually compare the variability of multiple datasets using a histogram or a dotplot.
Calculate variance and range given descriptive statistics.
Describe the variability of data using a data analysis tool.
Find and interpret the standard deviation of data.

The following summary of this section’s concepts and skills may be helpful to boost your confidence before trying the self-check.

Study Tips

Mnemonics & Symbols

• Visual comparisons
  • Data that’s spread wide over the graph has greater variability.
  • Tightly clustered, mounded in the middle has less variability.
• Range = the diffeRence between minimum and maximum values
• Be mindful of the letters used to represent standard deviation and mean in the formulas
  • Greek letters -> populations ([latex]\sigma[/latex] for standard deviation and [latex]\mu[/latex] for mean)
  • Latin letters -> samples ([latex]s[/latex] for standard deviation and [latex]\bar{x}[/latex] for mean)

Evidence-based strategies for learning

• Retrieval Practice: visual connections and contrasts associated with variability can be learned by quizzing yourself using flashcards and memory dumping (writing out everything you can remember onto a blank sheet of paper then comparing your output to a list of key concepts).
  • Keep a running stack of flashcards to use throughout the course as you prepare for high-stakes tests like the final exam.
  • Use Spaced Practice to quiz yourself with the flashcards in short (5 – 15 minute) bursts every few days.
• Gain understanding of standard deviation by paraphrasing the technical terminology out loud as though you are explaining it to a younger student.
  • If working in your study group, take turns paraphrasing and critiquing one another’s explanations.
  • If working alone, use the video feature on your phone to record yourself, then check your understanding against the answers and feedback to the section questions.
• Handwrite your notes for memory. Practice writing out the formulas and notation, pausing to pay attention to the letters you use for each:
  • The Latin letter s is used for samples.
  • The Greek letter [latex]\sigma[/latex] is used for populations.
  • This Latin / Greek system of representation is generally true: Latin letters for samples and Greek letters for populations.
    • Population: [latex]\mu[/latex] for mean, [latex]\sigma[/latex] for standard deviation
    • Sample:  [latex]\bar{x}[/latex] for mean, [latex]s[/latex] for standard deviation
• Use Interleaved Practice (mixing up similar problems) to obtain and calculate standard deviation, variance, and range in mixed problem sets rather than practicing one skill thoroughly before moving to the next. The similarities in the formulas and terminology can be confusing.
  • Variance is the radicand (the value under the square root) of the standard deviation.
  • Standard deviation is required to calculate variance.
  • As you practice calculating standard deviation and variance in the text and assignments, pause periodically to think about why you are doing each step, how you know what steps to use to make the calculations by hand and via technology, and how you can know if your answer is reasonable or correct.
• Use a Read, Pause & Write notetaking method to re-read the text and assignments in sections. Pause after each section to write (without looking at the text or questions) brief, descriptive, succinct notes. Compare what you have written from memory with the text afterwards to check your understanding.

Algebra & Arithmetic Refresher

• Number-Word Combinations

Key Equations

• Deviation from the mean

[latex]\left(x-\bar{x}\right)[/latex]
where [latex]\left(x\right)[/latex] is the observation in the dataset, and [latex]\left(\bar{x}\right)[/latex] is the sample mean.

• Standard deviation of a population

[latex]\sigma = \sqrt{\dfrac{\sum \left(x-\mu\right)^2}{n}}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\mu\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\mu\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

• Standard deviation of a sample

[latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\bar{x}\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\bar{x}\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

• Variance of a population

[latex]\sigma^{2}=\dfrac{\sum\left(x-\mu\right)^{2}}{n}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\mu\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\mu\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

• Variance of a sample

[latex]s^{2}=\dfrac{\sum\left(x-\bar{x}\right)^{2}}{n-1}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\bar{x}\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\bar{x}\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

Glossary

[latex]s[/latex]

the standard deviation of a sample of observations.

[latex]\sigma[/latex]

the standard deviation of a population of observations.

[latex]s^{2}[/latex]

the variation of a sample of observations.

[latex]\sigma^{2}[/latex]

the variance of a population of observations.

deviation from the mean

the distance between an observation ([latex]{x}[/latex]) in a dataset and the mean [latex]\left(\bar{x}\right)[/latex] of the dataset.

range

the maximum (or largest) value – the minimum (or smallest) value.

standard deviation

a measure of how spread out observations are from the mean.

variability

a measure of how dispersed (spread out) the data are. It is often referred to as the spread, or dispersion, of a dataset.

variance

the standard deviation squared.

Topic Complete – now test your understanding in the Self-Check.