Difficulty: Introductory | Prerequisites: Basic arithmetic, summation notation (Σ)
Big picture: Central tendency is one of the first things you learn in any statistics course because it answers the most basic question about a dataset: where is the middle? Before you can talk about spread, outliers, or inference, you need a single number that represents the "typical" value. This material forms the foundation for everything that follows in STAT 211, from probability distributions to hypothesis testing. If you are coming in cold, know that this section assumes you can read summation notation and work with ordered data.
Central tendency describes the centre of a dataset using a single representative value. The three main measures are the mean (arithmetic average), the median (middle value when sorted), and the mode (most frequent value). Each captures a different aspect of "typical," and each has situations where it is the best choice.
Sample mean (X̄)
The sum of all observations divided by the number of observations. Denoted X̄ ("x-bar") for a sample. In simple terms, this is what most people mean when they say "the average."
Median
The middle value of a dataset once all observations are sorted from smallest to largest. If there is an even number of observations, the median is the average of the two middle values. Think of it as the value that splits the sorted data into two equal halves.
Mode
The observation that appears most frequently in a dataset. A dataset can have one mode, more than one mode (bimodal, multimodal), or no mode at all if every value appears the same number of times. In simple terms, it is the most popular value.
Summation notation (Σ)
A compact way to write "add up all the values." ΣX means "sum every value of X." This notation appears constantly in statistics formulas.
The formula is: X̄ = ΣX / n
ΣX is the sum of all observed values
n is the number of observations (the sample size)
Example from Quiz 1: a civil engineer measures suspended solids X (parts per million) in river water over 10 weekdays. The observed values are 4, 2, 4, 8, 3, 6, 8, 2, 5, 8.
ΣX = 4 + 2 + 4 + 8 + 3 + 6 + 8 + 2 + 5 + 8 = 50
n = 10
X̄ = 50 / 10 = 5
Sort or scan the data and count how often each value appears.
The value with the highest frequency is the mode.
Example: in the dataset {4, 2, 4, 8, 3, 6, 8, 2, 5, 8}, the value 8 appears three times, which is more than any other value.
Mode = 8
If two values tie for the highest frequency, the dataset is bimodal.
Step 1: Sort the data in ascending order.
Sorted: 2, 2, 3, 4, 4, 5, 6, 8, 8, 8
Step 2: Count the observations. If n is odd, the median is the middle value. If n is even, the median is the average of the two middle values.
Here n = 10 (even), so the median sits between the 5th and 6th values.
Position of middle values: n/2 = 5th value, and (n/2 + 1) = 6th value
5th value = 4, 6th value = 5
Median = (4 + 5) / 2 = 4.5
A common exam task is completing a table with rows for X, X², and (X − X̄). The key relationships to know:
X² is each observation squared individually (not the sum squared)
(X − X̄) is called the deviation from the mean for each observation
ΣX = sum of all observations
ΣX² = sum of each squared observation (here, 302)
Σ(X − X̄) always equals 0 for any dataset, because the positive and negative deviations from the mean always cancel out
X | 4 | 2 | 4 | 8 | 3 | 6 | 8 | 2 | 5 | 8 |
|---|---|---|---|---|---|---|---|---|---|---|
X² | 16 | 4 | 16 | 64 | 9 | 36 | 64 | 4 | 25 | 64 |
X−X̄ | −1 | −3 | −1 | 3 | −2 | 1 | 3 | −3 | 0 | 3 |
ΣX = 50, ΣX² = 302, Σ(X − X̄) = 0
Sample mean: X̄ = ΣX / n
Median (even n): average of the values at positions n/2 and (n/2 + 1) in the sorted data
Mode: the value with the highest frequency
Water quality monitoring (as in the quiz scenario) relies on these measures daily. An engineer reporting the average concentration of suspended solids to a regulatory agency would use the mean, but if a few extreme readings from a storm event skewed the data, the median would give a more representative picture. The mode tells you the most commonly observed concentration, which can be useful for setting baseline expectations.
"The mean and median are the same thing." They coincide only when the data is perfectly symmetric. For skewed data they can differ substantially, and here they do: mean = 5, median = 4.5.
"If no value repeats, there is no mode." Technically correct for that dataset, but students sometimes confuse this with "the mode doesn't exist" in a broader sense. The mode is simply not useful when all values are equally frequent.
"Σ(X − X̄) measures how spread out the data is." It does not. It always equals zero. That is why we square the deviations (or take absolute values) to measure spread, which is covered in Part 2.
"To find the median, just pick the middle number." This only works when n is odd. For even n, you must average the two central values.
⚠️ You will almost certainly be asked to compute X̄, the median, and the mode from raw data. Show the formula and your working.
⚠️ The summation table (X, X², X − X̄) is a classic exam format. Remember that Σ(X − X̄) = 0 is a built-in check on your arithmetic.
⚠️ For median with even n, do not forget to average the two middle values. Forgetting this is one of the most common point losses on quizzes.
⚠️ Know when each measure is most appropriate: mean for symmetric data, median for skewed data, mode for categorical data.
True or false: The sum of all deviations from the mean, Σ(X − X̄), is always zero.
Fill in the blank: For a dataset with an even number of observations, the median is the ______ of the two middle values.
True or false: The mean is always a value that appears in the dataset.
Fill in the blank: If a dataset has values {3, 3, 5, 7, 7, 7}, the mode is ______.
True or false: The median is resistant to outliers, but the mean is not.
Answers: 1. True. 2. Average (arithmetic mean). 3. False (e.g., the mean of {2, 5} is 3.5). 4. 7. 5. True.
Q: Given the data 4, 2, 4, 8, 3, 6, 8, 2, 5, 8, compute the sample mean.
A: ΣX = 50, n = 10. X̄ = 50/10 = 5.
Q: Using the same data, find the median.
A: Sorted data: 2, 2, 3, 4, 4, 5, 6, 8, 8, 8. With n = 10, the median is the average of the 5th and 6th values: (4 + 5)/2 = 4.5.
Q: What is the mode of the dataset?
A: 8, because it appears three times, more than any other value.
Q: Why does Σ(X − X̄) always equal zero?
A: Because the mean is the balancing point of the data. Positive deviations (values above the mean) and negative deviations (values below the mean) always sum to zero by definition.
Q: A dataset has values {10, 10, 20, 30, 100}. Would you report the mean or the median as the "typical" value, and why?
A: The median (20) is more representative because the extreme value of 100 pulls the mean (34) upward. The median is resistant to outliers.
This material connects directly to measures of variability (variance, standard deviation, quartiles), which use the mean as their starting point. The concept of Σ(X − X̄) = 0 is precisely why variance squares the deviations. Later in STAT 211, the sample mean becomes the centrepiece of sampling distributions and confidence intervals.
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