Measures of Central Tendency: Mean, Median & Mode
Mean, median and mode — formulas, merits and demerits, which to use for missing values, what outliers destroy, and how skewness shifts the three apart.
7. Central Tendency — Definition & Scope
[T]
Definition
A measure of central tendency gives a single value that represents the whole data, and it describes the centre position of that data.
“It will give a single value about the data, and it talks about the centre position of the data. If you move the data from ascending to descending order or descending to ascending, it will give a single value out of that data — that value is either mean, median, or mode.”
The three metrics and where each applies
| Metric | Use with |
|---|---|
| Mean | Continuous / numerical variables |
| Median | Continuous / numerical variables |
| Mode | Categorical variables only |
Where this sits in the syllabus
The instructor previewed the descriptive statistics sequence as:
- Measure of central tendency ← this section
- Measure of variability / dispersion
- Measure of frequency distribution, histogram, and box plot
Aside on charts: “Chart is not my big favourite, but we’ll see. However, not many people are using pie chart.”
8. Mean
[T]
Formula
Mean μ = Sum of all values ÷ Number of values“It’s very simple.”
Merits
- Uses all the observations in the dataset — “unlike median and mode.”
Demerits
- Affected by extreme outliers. “If you have any outliers in your data, it will impact a lot.”
- Missing values must be filled first. “If we have any missing values, first you need to fill those missing values to calculate the mean.”
- Not suitable for categorical data. You cannot calculate the mean of “product category.”
Note on terminology
“Mean or average? Almost — I’ll say both are the same.”
9. Median
[T+S]
Definition
The middle-most observation in the data, once it has been arranged in ascending or descending order.
Case 1 — odd number of values
Data: 3, 5, 3, 8, 1, 7
Sorted: 1, 3, 5, 7, 8
n = 5 → position = (n + 1) / 2 = 6 / 2 = 3rd value
Median = 5Formula: the ((n + 1) / 2)-th value.
Case 2 — even number of values
Data: 5, 9, 7, 12
Sorted: 5, 7, 9, 12
n = 4 → n / 2 = 2 → take the 2nd and 3rd values
Median = (7 + 9) / 2 = 8Formula: the average of the (n/2)-th and ((n/2) + 1)-th values.
“If it is odd numbers, it’s very easy.”
Merits
- It is not affected by extreme values (outliers)
Demerits
- It is not based on all observations
10. Mode
[T]
The most frequently occurring value in the data.
Used primarily — and per the instructor, only — for categorical variables.
“Mode, you can use only for categorical variables.”
11. Using Mean and Median — Missing Value Rules
[S]
From the slide titled “Using mean and median”:
| Situation | Replace missing values with |
|---|---|
| Numeric variable, no outliers present | Mean |
| Numeric variable, outliers present | Median |
| Symmetric data (median = mean) | Either — both values are the same |
This rule connects directly to §12 (outliers wreck the mean) and §13 (in symmetric distributions mean = median = mode).
12. Outliers
[T]
Definition
An outlier is an observation that does not follow the usual pattern of the other observations.
The sales-vs-profit illustration
Set up a scatter plot with sales on the x-axis and profit on the y-axis. Each point answers: “for this level of sales, how much profit was made?”
Most points follow a visible pattern. Then one point sits far off it — very low sales, very high profit.
The business reading given in class:
“It means that you need to suspect that company. Better not invest. Maybe how they are showing the profit is by selling some of their fixed assets or something — not just because of real sales.”
What outliers damage — and what survives
| Statistic | Effect of an outlier |
|---|---|
| Mean | Badly distorted |
| Standard deviation | Badly distorted |
| Variance | Badly distorted |
| Median | Unaffected |
| Quartiles Q1, Q2, Q3 | Largely unaffected |
“If your data is having really outliers, then mean will impact, standard deviation will impact — everything will impact.”
Instructor’s guidance
“Make sure that your data is not having the outliers. And if any outliers are there, just [remove that] number from your population and calculate the standard deviation for the remaining samples.”
The bigger point
An outlier is rarely just a number to delete. It is usually a signal — a fault, a fraud, an anomaly, or an event that deserves investigation. See §29 for the full treatment.
13. Skewness & Distribution Shape
[S]
Source note: the instructor annotated these slides heavily, but the transcript for this portion was not captured. Content below is from the slides plus one class-chat message.
The reference case — symmetric distribution
The slide “Heights of People (cm)” shows a histogram of heights clustering around ~170 cm in a smooth bell shape, labelled:
Mean = Median = ModeThe three shapes
| Distribution | Relationship | Shape |
|---|---|---|
| Symmetrical | Mean = Median = Mode | Balanced bell curve |
| Right-skewed (positive skew) | Mean > Median > Mode | Long tail stretching right |
| Left-skewed (negative skew) | Mean < Median < Mode | Long tail stretching left |
The memory hook
“The mean goes towards the tail.” (raised in the class chat: “Mean goes towards the tail (Right or left) — for easy…”)
Why it works: the mean uses every observation, and the tail is exactly where the extreme values live — so the mean gets dragged in the tail’s direction. The mode stays at the peak (it’s the most frequent value). The median sits between them.
What the annotations emphasised
On the right-skewed chart, the instructor circled the long right tail and labelled it. That tail region is where the extreme values / outliers sit — it is the mechanism that pulls the mean away from the peak while the median resists.
The practical link
This is precisely why the missing-value rule in §11 works: in symmetric data mean and median coincide and either is safe; in skewed data with outliers, only the median is reliable.