Descriptive Statistics

The One Thing

Descriptive statistics organizes, summarizes, and presents observed data through tables, diagrams, graphs, and numerical measures. A complete summary considers both central tendency and dispersion so that data sets can be interpreted and compared appropriately.

Definitions and Results

• Data: Collected information, observations, or measurements used for analysis.
• Primary Data: Data collected directly by the investigator for a specific purpose.
• Secondary Data: Data obtained from existing records, reports, surveys, or other sources.
• Qualitative Data: Non-numerical data describing qualities or categories, such as colour or gender.
• Quantitative Data: Numerical data that can be counted or measured.
• Discrete Data: Numerical data consisting of countable values, usually whole numbers.
• Continuous Data: Numerical data that can take any value within an interval.
• Frequency: The number of times a particular value or class occurs.
• Frequency Distribution: A table showing values or class intervals together with their frequencies.
• Class Interval: A range of values used to group continuous or large data sets.
• Class Mark: The midpoint of a class interval:

$$\text{Class mark}=\frac{\text{lower limit}+\text{upper limit}}{2}.$$

• Cumulative Frequency: The running total of frequencies, calculated by the less-than or more-than method.
• Arithmetic Mean: The sum of all observations divided by the number of observations.
• Median: The middle value when observations are arranged in ascending or descending order.
• Mode: The value or class with the highest frequency.
• Range: The difference between the largest and smallest observations:

$$\text{Range}=\text{Largest observation}-\text{Smallest observation}.$$

• Variance: The average of the squared deviations from the mean.
• Standard Deviation: The positive square root of variance:

$$\sigma =\sqrt{{\sigma }^2}.$$

• Coefficient of Variation: A relative measure of dispersion:

$$CV=\frac{\sigma }{\bar{x}}\times 100.$$ A smaller coefficient of variation indicates greater consistency or lower relative variability.

• Quartiles: Values dividing ordered data into four equal parts: $Q_1$, $Q_2$, and $Q_3$.
• Interquartile Range: The difference between the third and first quartiles:

$$IQR=Q_3-Q_1.$$

• Histogram: A graph of continuous grouped data using adjoining rectangles.
• Frequency Polygon: A graph formed by joining points representing class marks and their corresponding frequencies.
• Ogive: A cumulative frequency curve used to locate measures such as the median and quartiles.
• Bar Graph: A graph representing categories with separated bars.
• Pie Chart: A circular diagram in which each sector represents a category; the sector angle is:

$$\frac{\text{category frequency}}{\text{total frequency}}\times 360^{\circ }.$$

The total of all frequencies equals the total number of observations. For continuous classes, class boundaries may need adjustment when intervals have gaps so that the intervals are continuous.

Mean uses every observation but can be strongly affected by extreme values. The median is more resistant to extreme values, while the mode is useful for identifying the most common category or value.

Worked Methods

1. Constructing and interpreting a frequency distribution

• Identify whether the data are qualitative, quantitative, discrete, or continuous.
• For large or continuous data sets, group observations into clear class intervals.
• Ensure that the classes are non-overlapping and cover all observations.
• Count the observations in each value or class to obtain its frequency.
• Add frequencies successively to obtain cumulative frequencies using either the less-than or more-than method.
• For each class interval, calculate the class mark:

$$x=\frac{\text{lower limit}+\text{upper limit}}{2}.$$

• Check that the total frequency equals the total number of observations.

2. Calculating the arithmetic mean

For ungrouped data:

• Add all observations to obtain $\Sigma x$.
• Count the observations to obtain $n$.
• Divide:

$$\bar{x}=\frac{\Sigma x}{n}.$$

For a frequency distribution:

• Multiply each value or class mark $x$ by its frequency $f$.
• Add the products to obtain $\Sigma fx$.
• Add the frequencies to obtain $\Sigma f$.
• Calculate:

$$\bar{x}=\frac{\Sigma fx}{\Sigma f}.$$

For grouped data using the assumed-mean method:

• Select an assumed mean $a$, usually a convenient class mark.
• Calculate the deviation:

$$d=x-a.$$

• Multiply each deviation by its frequency to obtain $fd$.
• Apply:

$$\bar{x}=a+\frac{\Sigma fd}{\Sigma f}.$$

For grouped data using the step-deviation method:

• Select an assumed mean $a$.
• Let $h$ be the class width.
• Calculate:

$$u=\frac{x-a}{h}.$$

• Multiply each $u$ by its frequency to obtain $fu$.
• Apply:

$$\bar{x}=a+h\left(\frac{\Sigma fu}{\Sigma f}\right).$$

3. Finding the median

For an individual series:

• Arrange the observations in ascending or descending order.
• If $n$ is odd, locate the observation in position:

$$\frac{n+1}{2}.$$

• If $n$ is even, take the average of the observations in positions:

$$\frac{n}{2}\text{and}\left(\frac{n}{2}+1\right).$$

For a continuous frequency distribution:

• Calculate the total frequency $N$.
• Find $N/2$.
• Use the cumulative frequencies to identify the median class.
• Record:

- $l$: lower boundary of the median class; - $cf$: cumulative frequency before the median class; - $f$: frequency of the median class; - $h$: class width.

• Apply:

$$\text{Median}=l+\left[\frac{N/2-cf}{f}\right]h.$$

4. Finding the mode

For a discrete distribution:

• Identify the value with the highest frequency.
• This value is the mode.

For a continuous frequency distribution:

• Identify the modal class, which has the greatest frequency.
• Let:

- $l$ be the lower boundary of the modal class; - $f_1$ be the modal-class frequency; - $f_0$ be the preceding frequency; - $f_2$ be the succeeding frequency; - $h$ be the class width.

• Apply:

$$\text{Mode}=l+\left[\frac{f_1-f_0}{2f_1-f_0-f_2}\right]h.$$

For a moderately skewed distribution, the empirical relation is: $$\text{Mode}\approx 3\text{Median}-2\text{Mean}.$$

5. Calculating range, variance, and standard deviation

To calculate the range:

• Identify the largest observation.
• Identify the smallest observation.
• Subtract:

$$\text{Range}=\text{Largest observation}-\text{Smallest observation}.$$

For an ungrouped population variance:

• Calculate the mean $\bar{x}$.
• Find each deviation $x-\bar{x}$.
• Square each deviation.
• Add the squared deviations.
• Divide by $n$:

$${\sigma }^2=\frac{\Sigma (x-\bar{x}{)}^2}{n}.$$

• Take the positive square root to obtain the standard deviation:

$$\sigma =\sqrt{{\sigma }^2}.$$

For a frequency distribution:

• Calculate $\bar{x}$.
• Find and square each deviation $x-\bar{x}$.
• Multiply each squared deviation by its frequency.
• Divide by the total frequency:

$${\sigma }^2=\frac{\Sigma f(x-\bar{x}{)}^2}{\Sigma f}.$$

An alternative variance formula is: $${\sigma }^2=\frac{\Sigma f{x}^2}{\Sigma f}-{\bar{x}}^2.$$

6. Calculating the coefficient of variation

• Calculate the standard deviation $\sigma$.
• Calculate the mean $\bar{x}$.
• Divide the standard deviation by the mean and multiply by $100$:

$$CV=\frac{\sigma }{\bar{x}}\times 100.$$

• When comparing data sets, the smaller coefficient of variation indicates greater consistency or lower relative variability. This relative measure is particularly suitable when data sets have different units or different means.

7. Representing data graphically

For a bar graph:

• Identify the categories.
• Represent each category with a separated bar.
• Use bar heights to show frequencies or values.

For a histogram:

• Group continuous data into class intervals.
• Place class intervals on the horizontal axis.
• Draw adjoining rectangles whose heights represent frequencies.

For a frequency polygon:

• Calculate the class mark for each class.
• Plot each class mark against its corresponding frequency.
• Join the plotted points with straight lines.

For an ogive:

• Calculate cumulative frequencies.
• Plot class boundaries against cumulative frequencies.
• Join the points to form a cumulative frequency curve.
• Use the curve to locate measures such as the median and quartiles.

For a pie chart:

• Find the total frequency.
• Calculate each category’s sector angle:

$$\frac{\text{category frequency}}{\text{total frequency}}\times 360^{\circ }.$$

• Draw sectors with the calculated angles.

Where It Goes Wrong

• Treating qualitative data, such as colour or gender, as though it were numerical quantitative data.
• Failing to distinguish discrete countable values from continuous measurements that can take any value within an interval.
• Constructing class intervals that overlap, leave gaps, or fail to cover all observations; continuous classes may require adjusted class boundaries.
• Forgetting that the total of all frequencies must equal the total number of observations.
• Confusing a bar graph, which uses separated bars for categories, with a histogram, which uses adjoining bars for continuous data.
• Interpreting a central value without considering dispersion, or forgetting that the mean is strongly affected by extreme values while the median is more resistant.

What Gets Asked

This material supports questions requiring students to:

• Define and distinguish primary, secondary, qualitative, quantitative, discrete, and continuous data.
• Construct or complete a frequency distribution, including class intervals, class marks, and cumulative frequencies.
• Calculate the arithmetic mean for ungrouped and frequency-distribution data.
• Apply the assumed-mean and step-deviation methods for grouped data.
• Find the median of an individual series and a continuous frequency distribution.
• Identify the mode or calculate the grouped-data mode.
• Use the empirical relation $\text{Mode}\approx 3\text{Median}-2\text{Mean}$ for moderately skewed distributions.
• Calculate the range, variance, standard deviation, quartiles, and interquartile range.
• Calculate and interpret the coefficient of variation when comparing variability.
• Draw or interpret bar graphs, histograms, frequency polygons, ogives, and pie charts.
• Explain why both central tendency and dispersion are needed to compare data sets.
• Distinguish descriptive statistical summaries from conclusions about cause and effect or predictions of future outcomes.