Mean Deviation in Frequency Distributions

A frequency table is a shorter way to write repeated data

A value recorded ten times should contribute ten times to a distance total. Frequency tables save us from writing every repeated observation separately. They do not change the underlying definition of mean deviation: we still average one distance for each original observation. Multiplying a value’s distance by its frequency accounts for all its repetitions at once.

If the table contains value 5 with frequency 8, those eight observations contribute 8 × 5 to the original-value total and 8 × |5 − a| to the absolute-deviation total about a. This is why frequencies must appear in both the centre calculation and the dispersion calculation. An unweighted average of the listed values would describe a different dataset.

Cumulative frequency locates a discrete median

A median is about ordered positions, not about averaging the listed values. Put the distinct values in ascending order and add a cumulative-frequency column. Each cumulative count tells us how many observations have been accounted for up to that value. It therefore identifies which value occupies any chosen position in the full dataset.

For odd N, locate position (N + 1)/2. For even N, locate both positions N/2 and N/2 + 1, then average the corresponding values. Both positions often belong to the same frequency row, but not always. If a cumulative count equals N/2 exactly, the next position belongs to the following row; simply choosing the first row reaching N/2 can miss this case.

Class intervals replace exact values by representatives

Sometimes a table records how many observations fall in intervals rather than their exact values. A continuous frequency distribution has adjacent class boundaries with no gaps. To calculate from such a table, represent each class by its midpoint. This is an approximation to the original data: we no longer know exactly where each observation lay within its class.

For intervals 10–20, 20–30 and so on, use midpoints 15, 25 and so on. Adopt a consistent boundary convention, such as including a lower boundary and excluding the upper boundary, so a boundary observation is counted once. The frequency still tells us how many original observations are represented by each midpoint.

Assumed means and step deviations simplify the mean

Large midpoint values can make multiplication tedious. An assumed mean A is a convenient reference value near the centre. Subtracting A shifts the origin; dividing by a positive common scale h reduces the numbers further. We then calculate a mean in the coded values and transform it back to the original unit.

Define dᵢ = (xᵢ − A)/h, so xᵢ = A + hdᵢ. Multiplying by fᵢ and summing gives Σfᵢxᵢ = AΣfᵢ + hΣfᵢdᵢ = AN + hΣfᵢdᵢ. Dividing by N gives the step-deviation mean formula. A is an assumed reference, not automatically the true mean.

Estimating a median within a class

For continuous grouped data we know which class contains the middle, but not the exact observation there. The grouped median formula estimates a position within that class by assuming observations are distributed evenly through its width. Cumulative frequencies identify the class first; interpolation then estimates the median itself.

Let the median class have lower boundary l, width h and frequency f. Let C be the cumulative frequency before it, and let total frequency be N. Of the N/2 observations needed to reach the middle, C have already occurred before the class. The remaining fraction (N/2 − C)/f of the class is converted to a distance by multiplying by h and adding l.

If N/2 is exactly a cumulative boundary, interpolation in either of the two adjacent continuous classes gives that shared boundary when the classes meet properly. Otherwise select the first class whose cumulative frequency exceeds N/2. Once M is estimated, use distances of class midpoints from M and weight them by frequency.

Make inclusive classes continuous when appropriate

An age table may use 16–20, 21–25 and so on for ages recorded to the nearest whole year. These written limits have apparent gaps, but the real dividing boundary between 20 and 21 is 20.5. For equally spaced whole-unit recording, subtract 0.5 from each lower limit and add 0.5 to each upper limit. The midpoints stay the same while the classes become adjacent.

What mean deviation can and cannot tell us

Mean deviation is an understandable average distance, but it remains only one number. It cannot reconstruct a distribution’s shape, identify which particular values were extreme, or guarantee that a chosen centre represents every subgroup. For grouped data its accuracy also depends on the midpoint and median-interpolation assumptions.

The median minimises the sum of absolute deviations; the mean generally does not. This explains why mean deviation about the median is no larger than mean deviation about the mean for the same raw data. It does not make either calculation meaningless. They describe distances from different centres and should be interpreted with those centres stated.

Absolute values can be handled mathematically, but their case-dependent signs make some algebraic manipulations less straightforward than those involving squares. Variance and standard deviation use squared deviations and lead to useful identities for shifting, scaling and correcting data. They complement the distance interpretation developed here.

Check your understanding

Track whether a table contains exact values or class intervals. For median questions, distinguish the discrete middle-position rule from continuous interpolation; for every average use the total frequency N.