Reverse and corrected summaries
Use the exact stated mean to recover a total. A rounded mean may not determine an exact total
- Missing value Subtract the known total from the required total \(\displaystyle x=n\bar{x}-T_{\mathrm{known}}\)
- Corrected value The count is unchanged \(\displaystyle T_{\mathrm{new}}=T-\mathrm{old}+\mathrm{new}\)
- Median correction Sort the corrected list before finding the central value or pair
- Underdetermined median An unknown away from the middle may have many possible values
- Added observation Add the new value to the total and increase the count by one. A correction, unlike an addition, keeps the count fixed
- Combined groups Add the group totals and divide by the combined count. Give larger groups more weight \(\displaystyle \bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}\)
Worked example
Five readings 4, 6, 8, 10, 22 have mean 10. Correct 22 to 32. Find the revised mean and median
- \(\displaystyle T=5\times10=50\)Recover the original total
- \(\displaystyle T_{\mathrm{new}}=50-22+32=60\)Replace one value without changing the count
- \(\displaystyle \bar{x}_{\mathrm{new}}=60/5=12\)Divide by the same five observations
- \(\displaystyle 4,6,8,10,32\Rightarrow\operatorname{median}=8\)The sorted central value remains 8.
Common mistakes
Difficulty levels
Questions about this topic
Can I multiply a rounded mean to get an exact total?+
Not generally. Rounding loses information. These missing-value questions state an exact mean
Does correcting a value change the count?+
No. It replaces one recorded value; adding a new observation is different
Does a correction always leave the median unchanged?+
No. Re-sort and check the central positions. The supplied extreme-correction practice is constructed to keep them unchanged
Can a median determine a missing value?+
Sometimes. An unknown in a central position can be constrained, but one outside it may have many possible values
Why can the mean lie between observed values?+
It divides the total equally among all observations and need not itself occur in the list
When can I average two group means directly?+
When the groups have equal counts, or when their means already agree. Otherwise recover and add the totals, then divide by the combined count
How should rounded answers be entered?+
Follow the precision stated in the question. For one decimal place, include one digit after the decimal point, even when it is zero. Keep exact totals until the final rounding step
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions