Measures of Central Tendency

What do you mean by “typical”?

Five orders take 2, 3, 3, 4, and 8 minutes to prepare. Their mean is 4 minutes, median is 3 minutes, and mode is 3 minutes. The values differ because each summary answers a different question.

• Mean: what equal value would preserve the total?
• Median: where is the middle of the ordered observations?
• Mode: which recorded value or category occurs most often?

Choose the measure for the question. Do not select it after seeing which answer is more flattering.

Calculate all three on the same data

Mean: use the total and the count

$\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}$

For the five waits, the total is 20 minutes and n = 5, so x̄ = 20/5 = 4 minutes. Replacing all waits by 4 would preserve their total.

Every observation contributes. If the last wait changes from 8 to 18, the mean rises to 30/5 = 6. That sensitivity can be useful: long waits really do increase the total time customers spend waiting. It can also make the mean a poor description of the middle customer.

Median: sort before choosing the middle

With an odd number of observations, choose the value at position (n + 1)/2. The third value in 2, 3, 3, 4, 8 is 3 minutes.

With an even number of numerical observations, average the two middle values. For 2, 3, 4, 8, the median is (3 + 4)/2 = 3.5 minutes. The median need not be an observed value.

The median is resistant to changes in the extreme values; it is not entirely unaffected by changes to a dataset. Adding or removing observations can move its middle position.

Mode: count frequencies

The value 3 appears twice; every other value appears once. The mode is 3 minutes. The mode also works for categories: if orders are tea, coffee, coffee, and juice, the modal category is coffee.

Ties can produce several modes. If every observed number is unique, introductory courses often say there is “no mode”; under a strict maximum-frequency definition, every value ties. State the convention if it matters. For continuous data, the most common exact recorded value can depend heavily on rounding.

Choose a summary that serves the question

Question	Useful starting measure	Why
What is the average cost per order?	Mean	Total cost divided by orders answers a budgeting question
What does the middle customer wait?	Median	Less driven by a few very long waits
Which product is ordered most?	Mode	Categories cannot be averaged numerically
What do the slowest 10% experience?	90th percentile	A measure of center misses the upper tail

A mean remains a valid summary of skewed data. It may simply answer a different question from the median. For customer experience, reporting both median and upper percentiles can be more informative than reporting the mean alone.

Worked example: combine groups correctly

Shift A has 10 orders averaging 4 minutes. Shift B has 30 orders averaging 6 minutes. The overall mean is not (4 + 6)/2 = 5.

Recover the totals first:

$\bar{x}_{combined}=\frac{10(4)+30(6)}{10+30}=\frac{220}{40}=5.5\text{ minutes}$

The larger shift gets more weight because it contributes more orders. An unweighted average of group means is appropriate only when the intended unit is a group or the groups have equal sizes.

This also explains why averaging percentages can fail. A success rate of 80% from 10 attempts and 50% from 100 attempts combine to (8 + 50)/(10 + 100) = 52.7%, not 65%.

Center needs context and spread

Datasets 3, 3, 3, 3, 3 and 1, 2, 3, 4, 5 both have mean and median 3. Their variability is very different. Pair a mean with standard deviation or a median with an interquartile range, and inspect a plot.

Mean greater than median often suggests a right tail, but mean–median–mode ordering is not a theorem for every distribution. A symmetric distribution can have two modes. Do not diagnose shape from three numbers alone.

Practice with explained solutions

1. Find the mean, median, and mode of 23, 29, 20, 32, 23, 21, 33, 25.

Show the calculation

The sum is 206, so the mean is 206/8 = 25.75. In order: 20, 21, 23, 23, 25, 29, 32, 33. The middle values are 23 and 25, so the median is 24. The mode is 23, the only repeated value.

2. Four salaries are 30, 35, 40, and 95 thousand dollars. What happens if 95 becomes 195?

Show the explanation

The mean rises from 200/4 = 50 to 300/4 = 75 thousand dollars. The median stays at (35 + 40)/2 = 37.5. The median describes the middle salaries; the mean describes total payroll per employee. Neither alone shows the full salary distribution.

Put it into practice

Paste the examples into the descriptive statistics calculator. Next, learn standard deviation and variance and percentiles and quartiles.