Source: Lecture notes / study guide
Difficulty: Introductory to Intermediate | Prerequisites: Variables, Measurement, and Sampling (Doc 1 of this set).
Once you have collected data and know what type of variable you are dealing with, the next step is to organise and describe it. Frequency distributions turn raw data into structured tables. Graphs turn those tables into pictures you can read at a glance. Measures of central tendency, variation, and position then reduce the data to a handful of numbers that capture its centre, spread, and shape. These descriptive tools are the foundation for every inferential method you will meet later: you cannot interpret a confidence interval or a test statistic without understanding what a mean, standard deviation, or z-score tells you.
Frequency distributions organise raw data into classes and counts. Graphs (histograms, polygons, ogives, bar charts, Pareto charts, pie charts, time-series plots) make patterns visible. Central tendency (mean, median, mode) locates the centre; variation (range, variance, standard deviation) measures the spread; and positional measures (z-scores, percentiles, quartiles, IQR, boxplots) tell you where individual values sit relative to the rest of the data.
Frequency distribution
An organisation of raw data into a table of classes and their frequencies. Think of it as a tally sheet that shows how often each category or range of values appears.
Class
A category (quantitative or qualitative) into which each raw data value is placed.
Class boundaries
Numbers used to separate classes so there are no gaps. Found by subtracting 0.5 from the lower class limit and adding 0.5 to the upper class limit.
Histogram
A graph of contiguous vertical bars whose heights represent the frequencies of interval or ratio data. Bars touch because the data is continuous.
Frequency polygon
A line graph connecting the midpoints of each class at their frequency. Useful for comparing two distributions on the same axes.
Ogive (cumulative frequency graph)
A graph that shows how many values fall below each upper class boundary. Think of it as a running total drawn as a curve.
Relative frequency
The frequency of a class divided by the total number of observations. All relative frequencies in a distribution sum to 1.
Mean
The sum of all values divided by the number of values. Think of it as the balance point of the distribution.
Median
The middle value when data is sorted from smallest to largest. It splits the data so that 50% of values fall at or below it.
Mode
The most frequently occurring value in a data set. A distribution can be unimodal (one mode), bimodal (two), or multimodal (more than two).
Range
The difference between the highest and lowest values. Quick but crude: it uses only two data points.
Variance
The average of the squared distances from the mean. It quantifies how spread out the data is, in squared units.
Standard deviation
The square root of the variance. It returns the spread measure to the original units of the data.
Coefficient of variation (CVar)
The standard deviation divided by the mean, expressed as a percentage. Allows comparison of variability across data sets with different units or scales.
Z-score (standard score)
The number of standard deviations a data value is above or below the mean. A z-score of 0 means the value equals the mean; positive means above, negative means below.
Percentile
The position of a value expressed as the percentage of data values that fall below it.
Quartiles (Q1, Q2, Q3)
Values that divide a sorted data set into four equal groups. Q1 = 25th percentile, Q2 = median = 50th percentile, Q3 = 75th percentile.
Interquartile range (IQR)
Q3 minus Q1. It captures the spread of the middle 50% of the data and is used to identify outliers.
Boxplot (box-and-whisker plot)
A graph showing the minimum, Q1, median, Q3, and maximum of a data set. Think of it as a five-number summary drawn as a picture.
A frequency distribution organises raw data into a table of classes and their counts. It reveals patterns, shows how data is distributed, and sets up further analysis (measures of average and spread, graphing).
Components of a frequency distribution table: class, tally, frequency. For quantitative data, add class limits and class boundaries.
Rules for constructing classes:
Use between 5 and 20 classes.
Class limits should have the same decimal place value as the raw data.
Class boundaries: lower limit minus 0.5 = lower boundary; upper limit plus 0.5 = upper boundary.
All classes must be equal in width. Width = range / number of classes. (Do not subtract limits within a single class; that gives the wrong answer.)
Classes must be mutually exclusive (no overlapping limits), continuous (no gaps), and exhaustive (every data point fits somewhere).
Sturge's Rule for choosing the number of classes: k = 1 + 3.322 × log₁₀(n), where k is the number of classes and n is the number of data points.
Types of frequency distribution:
Categorical: for nominal or ordinal data placed into specific categories.
Grouped: for quantitative data with a large range; classes are wider than one unit.
Ungrouped: for quantitative data with a small range; each class is a single value.
Cumulative: a running total of frequencies up to each upper class boundary (ascending cumulative frequency).
Graphs place class boundaries or categories on the x-axis and frequencies (or relative frequencies) on the y-axis.
Histogram: contiguous vertical bars for interval or ratio data. Bars touch because the underlying scale is continuous.
Frequency polygon: points plotted at the midpoint of each class at the class frequency, connected by straight lines. Good for comparing two distributions on the same axes.
Ogive (cumulative frequency graph): plots cumulative frequencies against upper class boundaries. Shows how many values fall below a given point.
Relative frequency graph: same shapes as above, but the y-axis shows proportions (each class frequency divided by the total). The sum of all relative frequencies is always 1.
Bar graph: vertical or horizontal bars representing frequencies of categorical data. Unlike histograms, bars do not touch.
Pareto chart: a bar chart for categorical data with bars arranged from tallest to shortest. Highlights the most common categories first.
Time-series graph: plots data points over time to reveal trends and patterns.
Pie graph: a circle divided into wedges proportional to frequencies. Best for nominal or categorical data when you want to show parts of a whole.
Constructing a graph: draw and label x and y axes, choose a suitable scale for the y-axis, place class boundaries (or categories) on the x-axis, plot points, and draw bars or lines.
Central tendency identifies the single value that best represents the centre of a distribution.
Mean (arithmetic average): sum of all values divided by the count. It is the "balance point" of the distribution. Sensitive to extreme values (outliers pull it toward the tail).
Median: the middle value of sorted data. It divides the distribution so that 50% of scores are at or below it. More resistant to outliers than the mean, making it the better measure when data is heavily skewed.
Mode: the most frequently occurring value. A data set can be unimodal (one mode), bimodal (two values tied for highest frequency), or multimodal (more than two). The modal class is the class with the highest frequency in grouped data.
Central tendency and distribution shape:
Symmetrical (normal) distribution: mean = median = mode, all at the centre.
Positively skewed (right-skewed): the tail extends to the right. The order from left to right is mode, median, mean. The mean is pulled toward the tail.
Negatively skewed (left-skewed): the tail extends to the left. The order from left to right is mean, median, mode.
A useful mnemonic: the mean always chases the tail.
Variation (or variability) describes how scattered the data is around the centre.
Range: highest value minus lowest value. Simple but uses only two data points, so it is easily distorted by a single outlier.
Variance: the average of the squared deviations from the mean. For a population, divide by N. For a sample, divide by n minus 1 (this correction, called Bessel's correction, produces an unbiased estimate of the population variance).
Standard deviation: the square root of the variance. Returns the measure to the original units, making it more interpretable than variance.
Coefficient of variation (CVar): standard deviation divided by the mean, times 100%. Useful for comparing variability between data sets measured in different units (e.g. comparing the spread of heights in centimetres to weights in kilograms).
Uses of variance and standard deviation:
Determine the spread of the data.
Assess the consistency of a variable.
Determine how many values fall within a specified interval (leads to the empirical rule and Chebyshev's theorem later).
Foundational to inferential statistics.
Positional measures tell you where a single data value sits relative to the rest of the distribution.
Z-score (standard score): z = (value minus mean) / standard deviation.
z = 0 means the value equals the mean.
Positive z means above the mean; negative z means below.
When all data values are converted to z-scores, the resulting distribution has a mean of 0 and a standard deviation of 1.
Percentile: the percentage of values that fall below a given data point. Formula: percentile = ((number of values below X) + 0.5) / (total number of values) × 100%.
Quartiles: divide the data into four equal parts.
Q1 = 25th percentile
Q2 = 50th percentile = median
Q3 = 75th percentile
Deciles divide the distribution into 10 equal parts (D1 = P10, D2 = P20, and so on). The median is P50 = Q2 = D5.
Interquartile range (IQR): Q3 minus Q1. It captures the middle 50% of the data and is a standard tool for detecting outliers.
Boxplot (box-and-whisker plot): a visual summary showing the five-number summary: minimum, Q1, median, Q3, maximum. The box spans Q1 to Q3 (the IQR), with a line at the median. Whiskers extend to the minimum and maximum. To compare two data sets, place their boxplots side by side: compare medians for location and box widths (IQR) for variability.
Stem-and-leaf plot: an exploratory tool that uses the leading digit(s) as the stem and the trailing digit as the leaf. Preserves the original data while showing the shape of the distribution.
Population variance: σ² = Σ(X − μ)² / N
Sample variance: s² = Σ(X − X̄)² / (n − 1)
Standard deviation: σ = √σ² (population) or s = √s² (sample)
Coefficient of variation: CVar = (s / X̄) × 100% (sample) or CVar = (σ / μ) × 100% (population)
Z-score: z = (value − mean) / standard deviation
Percentile: percentile = ((number of values below X) + 0.5) / (total number of values) × 100%
Sturge's Rule: k = 1 + 3.322 × log₁₀(n)
Class width: width = range / number of classes
Class boundaries: lower boundary = lower limit − 0.5; upper boundary = upper limit + 0.5
Histograms and frequency distributions are standard tools in quality control: manufacturing engineers use them to see whether a machine's output clusters within tolerance or drifts. Standard deviation is how financial analysts measure investment risk, since a fund with higher standard deviation has wider price swings. Z-scores underpin grading on a curve and are the basis for the normal-distribution calculations that appear in every field from psychology to engineering.
Students often subtract the limits of a single class to find class width. That gives the wrong answer. You must subtract the lower (or upper) limit of one class from the lower (or upper) limit of the next class.
A common error is treating the mean as always the best measure of centre. For skewed data, the median is more representative because the mean gets pulled toward the tail.
Students confuse variance and standard deviation. Variance is in squared units; standard deviation is in the original units. If your data is in kilograms, the variance is in kilograms-squared, which has no intuitive meaning, while the standard deviation is back in kilograms.
When reading a positively skewed distribution, students sometimes place the mean to the left. Remember: the mean chases the tail, so in a right-skewed distribution the mean is to the right of the median.
⚠️ Be able to construct a frequency distribution from raw data: choose the number of classes, compute class width, list limits and boundaries, and fill in frequencies.
⚠️ Know which graph matches which data type. Histograms and polygons are for quantitative (interval/ratio) data; bar charts and pie charts are for categorical data.
⚠️ Calculate mean, median, and mode, and identify which is appropriate given the shape of the distribution.
⚠️ Compute variance and standard deviation by hand for small data sets. Exams commonly give 5 to 10 values and ask for the sample standard deviation.
⚠️ Interpret z-scores: a z of 2.0 means the value is two standard deviations above the mean. Expect questions asking you to compare values from different distributions using z-scores.
⚠️ Draw and interpret boxplots, and use Q1, Q3, and IQR to identify the spread and potential outliers.
True or False: In a histogram, the bars do not touch. (False, histogram bars are contiguous)
Fill in the blank: The sum of all relative frequencies in a distribution equals ______. (1)
True or False: The median is more affected by outliers than the mean. (False, the mean is more affected)
Fill in the blank: The IQR is calculated as ______ minus ______. (Q3, Q1)
True or False: A z-score of negative 1.5 means the value is 1.5 standard deviations below the mean. (True)
Q: A data set has a range of 50 and you want 7 classes. What is the class width?
A: Class width = range / number of classes = 50 / 7 ≈ 7.14. Round up to 8 so all data is covered.
Q: In a right-skewed distribution, which measure of central tendency has the largest value: mean, median, or mode?
A: The mean. In a right-skewed distribution, the order from smallest to largest is mode, median, mean.
Q: A student scores 78 on an exam where the class mean is 70 and the standard deviation is 5. What is the student's z-score, and what does it mean?
A: z = (78 − 70) / 5 = 1.6. The student scored 1.6 standard deviations above the class mean.
Q: Given Q1 = 20 and Q3 = 45, what is the IQR? What range of values does it capture?
A: IQR = 45 − 20 = 25. It captures the middle 50% of the data.
Q: Why do we divide by n minus 1 (instead of n) when computing a sample variance?
A: Dividing by n minus 1 corrects for the fact that a sample tends to underestimate the population variance. This adjustment (Bessel's correction) produces an unbiased estimate of the population variance.
This connects to probability distributions because the shape of a frequency distribution (normal, skewed) directly informs which probability model applies. Z-scores are the gateway to the standard normal distribution and confidence intervals. Variance and standard deviation feed into hypothesis testing, where the test statistic is typically a ratio of an observed difference to a standard error. Boxplots and IQR connect to outlier detection methods used in exploratory data analysis.
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