Card 1 of 20
EasyStudy card 1
Prompt
What does standard deviation measure?
Think typical distance from center.
Answer
How spread out values are around their mean.
Example
Tight clustering means a smaller standard deviation.
MathChapter 8: Statistics
Flashcards
Standard Deviation Flashcards
Twenty concise cards on spread, calculation, transformations, normal intervals, and common traps.
Current card1 / 20
Session progress0 of 20 studied
Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What is a deviation from the mean?
Subtract the center from one value.
Answer
\(x_i-\bar{x}\).
Example
If \(x_i=9\) and \(\bar{x}=6\), the deviation is \(3\).
Card 3 of 20
EasyStudy card 3
Prompt
What population standard-deviation formula is used in this chapter?
Average squared deviations with denominator \(n\).
Answer
\(\sigma=\sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n}}\).
Example
Then take the principal square root.
Card 4 of 20
EasyStudy card 4
Prompt
Why square deviations?
Signed deviations sum to zero.
Answer
To prevent positive and negative deviations from canceling and measure magnitude.
Example
\((-2)^2=4\) and \(2^2=4\).
Card 5 of 20
EasyStudy card 5
Prompt
Can standard deviation be negative?
It is a square root of a nonnegative quantity.
Answer
No. It is always nonnegative.
Example
A result of \(-4\) cannot be a standard deviation.
Card 6 of 20
EasyStudy card 6
Prompt
When is standard deviation exactly \(0\)?
Every deviation must be zero.
Answer
When every value is identical.
Example
\(5,5,5\) has standard deviation \(0\).
Card 7 of 20
EasyStudy card 7
Prompt
What does a smaller standard deviation indicate?
Compare horizontal spread.
Answer
Values are more tightly clustered around the mean.
Example
\(9,10,11\) is tighter than \(0,10,20\).
Card 8 of 20
MediumStudy card 8
Prompt
Does a larger mean force a larger standard deviation?
Compare deviations from each set's own mean.
Answer
No. Center and spread are different properties.
Example
A set centered at \(100\) can have zero spread.
Card 9 of 20
MediumStudy card 9
Prompt
What happens to standard deviation after adding \(k\) to every value?
A shift preserves distances.
Answer
It remains unchanged.
Example
Adding \(10\) moves the mean but not the spread.
Card 10 of 20
MediumStudy card 10
Prompt
What happens after subtracting \(k\) from every value?
Subtraction is also a shift.
Answer
Standard deviation remains unchanged.
Example
Every deviation from the new mean matches its old deviation.
Card 11 of 20
MediumStudy card 11
Prompt
What happens after multiplying every value by positive \(k\)?
Scale every distance.
Answer
Standard deviation is multiplied by \(k\).
Example
Tripling the data triples standard deviation.
Card 12 of 20
MediumStudy card 12
Prompt
For \(y=kx+b\) with positive \(k\), what is the new standard deviation?
The shift \(b\) does not affect spread.
Answer
\(k\sigma_x\).
Example
If \(k=4\) and \(\sigma_x=3\), then \(\sigma_y=12\).
Card 13 of 20
MediumStudy card 13
Prompt
About what percent of a normal distribution is within \(1\) standard deviation?
Use the chapter-level normal approximation.
Answer
Approximately \(68\%\).
Example
From \(\mu-\sigma\) to \(\mu+\sigma\).
Card 14 of 20
MediumStudy card 14
Prompt
About what percent is within \(2\) standard deviations?
Use the middle interval.
Answer
Approximately \(95\%\).
Example
From \(\mu-2\sigma\) to \(\mu+2\sigma\).
Card 15 of 20
MediumStudy card 15
Prompt
About what percent is within \(3\) standard deviations?
Use the widest source-level interval.
Answer
Approximately \(99\%\).
Example
From \(\mu-3\sigma\) to \(\mu+3\sigma\).
Card 16 of 20
MediumStudy card 16
Prompt
About what percent lies between the mean and \(1\) standard deviation above it?
Half of the central \(68\%\).
Answer
Approximately \(34\%\).
Example
Symmetry splits the center equally.
Card 17 of 20
MediumStudy card 17
Prompt
About what percent lies above \(\mu+\sigma\)?
Outside the middle \(68\%\), split the remaining \(32\%\).
Answer
Approximately \(16\%\).
Example
The upper tail is about \(16\%\).
Card 18 of 20
HardStudy card 18
Prompt
When may the \(68/95/99\) approximation be used?
Do not assume every distribution is bell-shaped.
Answer
When the distribution is stated or shown to be approximately normal.
Example
A strongly skewed histogram does not qualify automatically.
Card 19 of 20
HardStudy card 19
Prompt
What is a common graph-comparison trap?
Standard deviation concerns dispersion along the value axis.
Answer
Comparing bar height instead of horizontal spread.
Example
A tall narrow graph can have less spread than a short wide graph.
Card 20 of 20
HardStudy card 20
Prompt
What is a fast transformation strategy?
Analyze \(kx+b\) in two parts.
Answer
Separate shifts from scales: shifts preserve spread; positive scales multiply it.
Example
For \(3x+7\), multiply standard deviation by \(3\) and ignore \(7\).
Deck complete
You rated every card. Revisit difficult cards or shuffle for another pass.
Keyboard: Space reveals, left/right arrows navigate, 1–4 rate, and S shuffles.