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1.6 · Calculate and interpret measures of centre for ungrouped data

Learn to calculate and interpret measures of centre for ungrouped data through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

Calculate and interpret the mean, median, and mode

A data set often needs a concise description of what is typical. A measure of centre gives one value that represents a central or common feature of the observations. This lesson covers the mean, median, and mode. They describe centre in different ways, so the useful choice depends on the variable and the values observed. Ungrouped data are individual observations listed separately rather than summarized into intervals.

What you will learn

  • Distinguish a population, sample, variable, parameter, and statistic.
  • Calculate the mean, median, and mode for ungrouped data.
  • Choose and interpret a measure of centre in context, including its units.

1. Data language and preparation

A population is the full group a study aims to describe. A sample is the part of that group from which data are collected. A variable is a characteristic recorded for each person or item, such as study time or number of books borrowed. A parameter is a number describing a feature of a population. A statistic is a number calculated from sample data. For example, the mean study time for every student in a program is a parameter; the mean for 30 surveyed students is a statistic.
Before calculating a measure of centre, identify what each observation represents and check that the values are recorded consistently. Numerical measures such as the mean and median require numerical data. For a category such as a student’s program, arithmetic is not meaningful, but the most common category can be found using the mode.
A brief algebra review helps with the mean: add the observations, then divide by their count. The symbol nn denotes the number of observations, and xix_i denotes the value in position ii. The notation ∑i=1nxi\sum_{i=1}^{n}x_i means to add the values from the first observation through the nnth. To find the median, arrange the observations from smallest to largest first; their original order does not determine the middle.
  • Ungrouped data are individual values, not values summarized into intervals.
  • Use consistent units and include each observation once.
  • Sort the observations before finding the median.

2. Mean: the arithmetic average

The mean is the sum of all observed values divided by the number of observations. For sample data, it is written as xˉ\bar{x} and pronounced “x-bar.” The mean has the same units as the variable. It uses every observation, so a very large or very small value can pull it toward itself.
For a sample, check that the observations are numerical, that they use the same units, and that you know how many values there are. Add the values, divide their sum by their count, and interpret the result in context. The arithmetic works for a finite list of numerical observations; whether the mean is a useful description of what is typical also depends on the values.
A mean calculated from a sample is a statistic. If the entire population is measured, its mean is a parameter, commonly written as μ\mu. The calculation follows the same idea, but this lesson’s example uses a sample mean.
xˉ=∑i=1nxin\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}
  • Mean equals the sum of the observations divided by their number.
  • A mean need not be one of the observed values.
  • An unusually high or low observation can affect the mean.

3. Median: the middle after sorting

The median is the middle value in observations arranged from least to greatest. It is a positional measure: the sorted positions determine it. For numerical observations, the median has the same units as the data and is generally less affected by an extreme value than the mean.
If the number of observations is odd, the median is the one value in the middle position. If the number is even, there are two middle values, and the median is their arithmetic mean. The notation x(j)x_{(j)} means the value in position jj after the data have been sorted. Thus, the median position for an odd count is (n+1)/2(n+1)/2; for an even count, average the values in positions n/2n/2 and n/2+1n/2+1.
Averaging two middle observations does not add a new observation to the data. It identifies the point halfway between those two values. For example, an even-sample median for whole-number counts may be halfway between possible observed counts; it is still a valid description of the centre.
Median={x((n+1)/2),n oddx(n/2)+x(n/2+1)2,n even\text{Median}=\begin{cases}x_{((n+1)/2)},&n\text{ odd}\\[2pt]\frac{x_{(n/2)}+x_{(n/2+1)}}{2},&n\text{ even}\end{cases}
  • Sort the observations before locating the middle.
  • With an odd count, use the one middle value; with an even count, average the two middle values.
  • The median is based on position and is generally less affected by extreme values than the mean.

4. Mode and choosing a useful centre

The mode is the value or category that occurs most often. To find it, count how many times each distinct value or category appears, then identify the greatest frequency. The mode can be used for numerical or categorical data, and it is always an observed value or category.
If one distinct value occurs more often than every other distinct value, it is the unique mode. If two or more distinct values tie for the greatest frequency, the data have multiple modes. If there are multiple distinct values and all occur equally often, there is no mode because none occurs more often than the others. In contrast, if every observation is identical, that shared value is the unique mode: it occurs at every position and is the only distinct value.
A measure of centre summarizes data; it does not claim that every observation is close to the summary. For numerical data without unusually distant values, the mean can be useful because it includes every observation. If a value is unusually far from the others, the median may better represent a typical observation. For categories, use the mode; arithmetic on category labels is not meaningful. Always interpret the measure by naming the variable, units when applicable, and whether the result describes a sample or a population.
  • The mode is the most frequent value or category.
  • Distinct values tied for the greatest frequency give multiple modes; if all distinct values have equal frequency, there is no mode.
  • When all observations are identical, their shared value is the unique mode.
  • Match the measure to the data type and observed values.

Worked example

Study time in a small sample

Five students report how many hours they studied last week: 2, 3, 3, 4, and 8. Find and interpret the mean, median, and mode.
  1. Identify the data
    The population of interest might be all students in a program, while these five respondents form a sample. The variable is reported study time in hours. The observations are numerical, use the same unit, and are listed individually, so the mean, median, and mode can be calculated. Results from these five students are statistics, not population parameters.
  2. Calculate the mean
    Add the five reported times and divide by the number of students. The substitution shows both the sum and the count, so repeated values are included as separate observations.
    xˉ=2+3+3+4+85=205=4 hours\bar{x}=\frac{2+3+3+4+8}{5}=\frac{20}{5}=4\text{ hours}
  3. Calculate the median
    The observations are already arranged from least to greatest. Because there are five observations, the median is the third value, the one in the middle position.
    Median=x((5+1)/2)=x(3)=3 hours\text{Median}=x_{((5+1)/2)}=x_{(3)}=3\text{ hours}
  4. Find the mode
    The value 3 occurs twice; each other value occurs once. Therefore, 3 hours is the unique most frequent reported study time.
    Mode=3 hours\text{Mode}=3\text{ hours}
  5. Interpret the results
    The sample mean study time is 4 hours, while the median and mode are both 3 hours. The 8-hour observation raises the mean above the middle and most common value. In this sample, 3 hours describes both the middle report and the most frequent report; 4 hours is the arithmetic average. Neither result means that every student studied for that amount.
Answer: Mean: 4 hours; median: 3 hours; mode: 3 hours.
Check: The observations total 20 hours, and dividing by 5 gives a mean of 4 hours. The third sorted observation is 3 hours, and 3 occurs more often than any other value.

Common mistakes and how to avoid them

Finding the median from the unsorted list or choosing the lower of the two middle values when the count is even.
Correction: Sort all observations first. With an even number of values, average the two middle values.
Dividing by the number of distinct values instead of the number of observations when finding the mean.
Correction: Count every observation, including repeats. The denominator is the total number of observations.
Assuming every data set has exactly one mode, or assuming identical observations have no mode.
Correction: Compare frequencies. Distinct values can tie for the greatest frequency, and multiple distinct values with equal frequency have no mode. If all observations are identical, their shared value is the unique mode.
Treating the mean as the value that most people recorded.
Correction: The mean is an arithmetic average. The mode is most frequent, and the median is the middle after sorting.
Reporting a centre without units or context.
Correction: Name the variable and its units when applicable, and say whether the result describes the sample or population.

Lesson summary

  • For numerical ungrouped data, calculate the mean by dividing the sum by the number of observations.
  • Find the median by sorting the data and locating the middle; average the middle pair when the count is even.
  • Find the mode by identifying the value or category with the greatest frequency. If all observations are identical, their shared value is the unique mode.
  • Interpret the measure in context. The mean can be pulled by extreme values, the median is based on position, and the mode is based on frequency.

Check your understanding

Question 1

For the values 1, 4, 4, and 7, what is the median?
  1. 4
  2. 3.5
  3. 4.5
  4. 7
Show answer and explanation
4
There are four sorted values, so average the two middle values: 4+42=4\frac{4+4}{2}=4.

Question 2

A set of responses is the categories red, blue, red, and green. Which measure of centre can identify the most common category?
  1. The mean
  2. The median
  3. The mode
  4. The sum
Show answer and explanation
The mode
Red appears twice, more often than either other category, so red is the mode. Averaging category names is not meaningful.

Question 3

A numerical data set contains one value far above the others. Which measure is generally less affected by that value?
  1. The mean
  2. The median
  3. The sum
  4. The number of observations
Show answer and explanation
The median
The median depends on the middle position after sorting, so a distant value generally affects it less than it affects the mean.

Key terms

Ungrouped data
Individual observations listed separately rather than summarized into intervals.
Population
The full group a study aims to describe.
Sample
The part of a population from which data are collected.
Variable
A characteristic recorded for each person or item in a study.
Parameter
A number describing a feature of a population.
Statistic
A number calculated from sample data.
Mean
The sum of the observations divided by their number.
Median
The middle value after observations are arranged in order, or the average of the two middle values.

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