Statistics helps us describe variation in observations and make careful claims about data. Before calculating, ask how information was collected, what each number represents and whether the observations are a whole population or a sample.
A correct arithmetic result can still be misleading if its context is ignored. The original examples here focus on common descriptive methods. Consult the current official STA111 course material for the full unit coverage, including any probability topics assessed in your programme.
What you will be able to do
- Recognise nominal, ordinal and quantitative variables.
- Construct and interpret a simple frequency table.
- Calculate and compare mean, median and mode.
- Find range and population variance for a small data set.
Lessons with worked examples
1. Classify data before choosing a summary
Categorical variables identify groups, while quantitative variables record numerical measurements or counts. Nominal categories have no intrinsic order, such as device type; ordinal categories have a meaningful order, such as poor, fair and good. Quantitative variables may be discrete counts or continuous measurements.
The classification influences the summary. It is not meaningful to calculate the arithmetic mean of labels such as “red” and “blue.” For an ordered rating, the median category may be more interpretable than an arithmetic calculation on arbitrary codes. Check what the data really measure before constructing charts.
- Separate counts from measurements.
- Avoid treating identification numbers as quantities.
- Choose a descriptive statistic suited to the variable.
Worked example — phone ownership survey
A class records phone operating system, hours of study per week and number of registered courses. Operating system is nominal categorical; hours of study is quantitative (often treated as continuous); the number of courses is a discrete count. A frequency table is useful for operating system, while a mean and distribution may help summarize study hours.
Takeaway: Data type comes before formula choice.
2. Frequency tables and trustworthy charts
A frequency table counts observations in categories or intervals. Relative frequency equals category frequency divided by the total number of observations; percentages multiply that fraction by 100. Check that all frequencies add up to the sample size and that non-overlapping classes cover the intended values.
Bar charts compare categories; histograms commonly summarize numerical values grouped into adjacent intervals. Axis labels and sensible scales are essential. A truncated vertical axis may exaggerate small differences; unequal class widths require care with histogram heights, which represent frequency density rather than raw count.
- Check the total frequency against sample size.
- Label charts with variable names and units.
- Look for misleading axis choices or omitted categories.
Worked example — revision hours
In a group of ten learners, 2 report fewer than 2 study hours daily, 5 report 2–4 hours and 3 report more than 4 hours. Relative frequencies are 2/10 = 20%, 5/10 = 50% and 3/10 = 30%. The total is 100%, confirming the reported classes cover the ten learners.
Takeaway: Relative frequency allows comparison across groups of different sizes.
3. Measures of centre: mean, median and mode
The arithmetic mean sums all observations and divides by the number of observations. The median is the middle value after sorting, or the average of the two middle values for an even-sized list. The mode is the most frequently occurring value; some data sets have multiple modes or no repeated value.
No single measure is best in every situation. An extreme outlier can pull the mean far from most observations, while the median is usually more resistant. When describing salaries or waiting times, report more than one measure and consider the shape of the data before claiming what is “typical.”
- Sort data before finding the median.
- Use the mean only for appropriate numerical variables.
- Explain outlier sensitivity when interpreting averages.
Worked example — five test scores
For 2, 3, 3, 4, 18: mean = (2+3+3+4+18)/5 = 30/5 = 6. Median is 3, and mode is 3. The mean is considerably higher than most scores because of the outlier 18. A report should mention that difference rather than declaring 6 typical without qualification.
Takeaway: Compare centre measures before interpreting a skewed distribution.
4. Spread: range, variance and standard deviation
Range is maximum minus minimum. Population variance averages the squared deviations from the population mean; population standard deviation is the square root of population variance. Squaring prevents positive and negative deviations from cancelling, while taking the square root returns to the original data units.
For a sample used to estimate population variance, the conventional sample-variance denominator is n−1 rather than n. State which formula you are using. Standard deviation shows typical scale of variation, not an absolute guarantee that every observation falls within a chosen distance of the mean.
- Compute deviations after calculating the mean.
- Distinguish population variance from sample variance.
- Interpret standard deviation in original data units.
Worked example — complete four-score population
Values: 2, 4, 6, 8. Population mean is 5. Deviations are −3, −1, 1 and 3; squared deviations sum to 9+1+1+9 = 20. Population variance = 20/4 = 5 score². Standard deviation = √5 ≈ 2.24 score units. If these four values were a sample, sample variance would instead be 20/3 ≈ 6.67.
Takeaway: The denominator depends on whether you are describing a population or estimating from a sample.
Apply what you learned — STA111 Practice Clinic
Original independent learning exercise. Try it before opening the worked explanation; this is not an official NOUN examination question.
Understand why the mean may mislead
Situation: Five learners have weekly study times of 3, 3, 6, 8 and 20 hours.
Your task: Calculate the arithmetic mean, median and range. Decide which measure better represents a typical learner in this small group.
Reveal the step-by-step solution
- Total study time = 3 + 3 + 6 + 8 + 20 = 40 hours.
- Mean = 40 ÷ 5 = 8 hours.
- The ordered middle (third) value is 6, so the median is 6 hours.
- Range = 20 − 3 = 17 hours; the large 20-hour value pulls the mean above the median.
Answer: Mean: 8 hours; median: 6 hours; range: 17 hours. The median better describes the middle learner here.
Common mistake to avoid: Always explain what “typical” means; the mean remains mathematically correct even when it is influenced by an extreme observation.
Test yourself — original practice questions
These are independently created learning questions, not official NOUN TMA or examination questions. Try each one before revealing the explanation.
Question 1. Which variable is nominal categorical?
- Height in centimetres
- Phone operating system
- Number of books owned
- Daily travel distance
Show answer and explanation
Answer B: Phone operating system
Operating system identifies categories that have no inherent numerical order.
Question 2. Three of twelve students choose a given option. Its relative frequency is:
- 12%
- 20%
- 25%
- 36%
Show answer and explanation
Answer C: 25%
Relative frequency = 3/12 = 0.25 = 25%.
Question 3. For data 1, 2, 2, 3, 12, what is the median?
- 2
- 3
- 4
- 12
Show answer and explanation
Answer A: 2
The sorted middle (third) observation is 2.
Question 4. For the population 2, 4, 6, 8, what is population variance?
- 4
- 5
- 6.67
- 20
Show answer and explanation
Answer B: 5
Squared deviations from mean 5 sum to 20; divide by population size 4 to get 5.
Three things to practise this week
- Collect ten observations of a non-sensitive variable and create a correctly labelled frequency table.
- Find the mean, median and mode for two short data sets, one containing an outlier.
- Calculate population variance for four values, then explain how sample variance would differ.
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