Grouped mean is a weighted midpoint average
Each class is represented by its class mark xi and weighted by frequency fi. The denominator is total frequency, not the number of classes.
Learning goals
- Calculate grouped mean using the direct method.
- Calculate and compare assumed-mean and step-deviation methods.
- Recover a missing frequency from a stated mean.
Concept lesson
Class marks and totals
Calculate xi=(lower+upper)/2 and N=sum fi. Check equal class width before using step deviation.
Signed weighted sums
Preserve signs when multiplying fi by deviations and include an unknown frequency symbol in both its product and total-frequency expressions.
Direct method
Build columns class interval, fi, xi and fixi. Then mean=sum fixi divided by sum fi. Estimate the centre first to catch unreasonable arithmetic.
Assumed mean and step deviation
Choose central a; use di=xi-a and mean=a+sum fidi/N. For equal width h, ui=di/h and mean=a+h(sum fiui/N). These transform arithmetic without changing the weighted mean.
Direct-method table
Calculate every midpoint and weighted product row, total both fi and fixi, divide once, and check that the result lies inside the overall data range.
Select the shortest method
Use direct method for small midpoint products, assumed mean for large midpoints near a convenient centre, and step deviation when equal widths make deviations share a common factor.
Missing frequency from known mean
If frequency k has class mark xk, write total frequency=N_known+k and weighted total=S_known+kxk. Set given mean=(S_known+kxk)/(N_known+k), then solve and validate k as non-negative.
Denominator and product audit
Check every fi, xi and fixi row, confirm N=sum fi, and estimate whether heavily weighted classes pull the mean toward their midpoints.
Deviation-column consistency
Choose a, calculate all signed di or ui values with one rule, complete weighted products, and justify the selected shortcut from number size and class width.
Inverse-mean equation audit
Highlight the unknown frequency in its table row, numerator term and denominator term; solve, require an admissible frequency and substitute it back into the mean.
Three-method invariance check
Compute one table using two methods, compare identical results, locate any discrepancy at midpoint, deviation, weighted product or final scaling, and recommend the efficient method.
Mean working-table protocol
Show labelled columns and totals, formula substitution and final value. For a missing frequency, show the unknown in both numerator and denominator and verify it.
Grouped-mean mastery
Calculate a direct mean, complete a shortcut-method table, justify method choice and solve an inverse missing-frequency item.