Card 1 of 20
EasyStudy card 1
Prompt
What is the arithmetic mean?
Think total per observation.
Answer
The sum of all values divided by the number of values.
Example
For \(3,7,8\), the mean is \(18/3=6\).
MathChapter 8: Statistics
Flashcards
Mean, Median, Mode, and Range Flashcards
Twenty concise cards on measures of center and spread, frequency tables, weighted groups, transformations, and traps.
Current card1 / 20
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
How do you recover a data total from its mean?
Reverse the division in the mean formula.
Answer
Multiply the mean by the number of observations: \(\sum x_i=n\bar{x}\).
Example
A mean of \(12\) for \(7\) values gives total \(84\).
Card 3 of 20
EasyStudy card 3
Prompt
What must you do before finding a median?
Median is based on position.
Answer
Order the data values from least to greatest.
Example
Order \(9,2,6\) as \(2,6,9\).
Card 4 of 20
EasyStudy card 4
Prompt
How is the median found when the count is odd?
Use position \((n+1)/2\).
Answer
Select the single middle value in the ordered list.
Example
For \(n=7\), use the fourth value.
Card 5 of 20
EasyStudy card 5
Prompt
How is the median found when the count is even?
Use positions \(n/2\) and \(n/2+1\).
Answer
Average the two middle values in the ordered list.
Example
The middle values \(8\) and \(12\) give median \(10\).
Card 6 of 20
MediumStudy card 6
Prompt
What is the mode?
Mode measures repetition, not size.
Answer
The value or values with the greatest frequency.
Example
In \(2,2,5,9\), the mode is \(2\).
Card 7 of 20
MediumStudy card 7
Prompt
Can a data set have more than one mode?
Compare all frequencies.
Answer
Yes. Every value tied for greatest frequency is a mode.
Example
In \(1,1,4,4,7\), both \(1\) and \(4\) are modes.
Card 8 of 20
MediumStudy card 8
Prompt
When does a data set have no mode?
Distinct values have equal frequency \(1\).
Answer
When no value occurs more often than another.
Example
The set \(3,6,9\) has no mode.
Card 9 of 20
EasyStudy card 9
Prompt
What is the range?
Use only the endpoints.
Answer
Maximum minus minimum.
Example
For \(5,8,17\), range \(=17-5=12\).
Card 10 of 20
MediumStudy card 10
Prompt
How do you find a missing value when the mean is known?
Required total is \(n\bar{x}\).
Answer
Compute the required total, then subtract the sum of the known values.
Example
Four values with mean \(10\) total \(40\).
Card 11 of 20
MediumStudy card 11
Prompt
What is the frequency-table mean formula?
Weight each value by its frequency.
Answer
\(\bar{x}=\dfrac{\sum xf}{\sum f}\).
Example
Value \(5\) with frequency \(3\) contributes \(15\).
Card 12 of 20
MediumStudy card 12
Prompt
How are two group means combined?
Weight means by group size.
Answer
Convert each mean to a total, add totals, and divide by the combined count.
Example
Use \((n_1m_1+n_2m_2)/(n_1+n_2)\).
Card 13 of 20
MediumStudy card 13
Prompt
Why can averaging two group means be wrong?
Equal weighting is justified only for equal group sizes.
Answer
The groups may have different numbers of observations.
Example
Means \(70\) and \(80\) do not force combined mean \(75\).
Card 14 of 20
MediumStudy card 14
Prompt
What happens to the mean when \(k\) is added to every value?
The whole distribution shifts.
Answer
The mean increases by \(k\).
Example
Adding \(4\) changes mean \(9\) to \(13\).
Card 15 of 20
MediumStudy card 15
Prompt
What happens to the range when \(k\) is added to every value?
Both endpoints shift equally.
Answer
The range is unchanged.
Example
\((b+k)-(a+k)=b-a\).
Card 16 of 20
MediumStudy card 16
Prompt
What happens when every value is multiplied by positive \(k\)?
A positive scale factor preserves order.
Answer
The mean, median, mode values, and range are multiplied by \(k\).
Example
Doubling the data doubles these measures.
Card 17 of 20
MediumStudy card 17
Prompt
Which measure of center is generally more resistant to an extreme outlier?
It depends on position more than magnitude.
Answer
The median.
Example
One huge maximum can pull the mean upward while the middle position barely changes.
Card 18 of 20
MediumStudy card 18
Prompt
Which summary is directly determined by only the minimum and maximum?
Ignore interior values for this measure.
Answer
The range.
Example
Changing an interior value may leave the range unchanged.
Card 19 of 20
HardStudy card 19
Prompt
How do you update a mean after adding a value \(x\)?
Update total and count.
Answer
Use \((n\bar{x}+x)/(n+1)\).
Example
Five values with mean \(8\), plus \(14\), give \(54/6=9\).
Card 20 of 20
HardStudy card 20
Prompt
What is a reliable SAT check for an ordinary mean?
A weighted balance cannot fall outside all data values.
Answer
The mean should lie between the minimum and maximum.
Example
A claimed mean of \(20\) for values from \(4\) to \(12\) is impossible.
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