01. What is statistics?
Statistics is the study of collecting, organizing, analyzing, interpreting, and presenting data. It helps us discover patterns without assuming that every observation tells the whole story.
Two important branches
Descriptive statistics summarizes the data we already have, using measures such as the mean, median, range, and standard deviation.
Inferential statistics uses a sample to make cautious estimates or conclusions about a larger population.
For example, calculating the average score of one class is descriptive statistics. Using a carefully selected sample of students to estimate a wider trend is inferential statistics.
02. Mean, median, and mode
These measures describe the centre of a dataset, but they do not always describe it in the same way.
Mean — the arithmetic average
Add all observations and divide by the number of observations.
Data: \(2,4,4,6,9\)
\[ \bar{x}=\frac{2+4+4+6+9}{5}=\frac{25}{5}=5 \]
The mean is \(5\).
Median — the middle value
Arrange the values in order. If there is an odd number of observations, choose the middle one. If there is an even number, average the two middle values.
This position formula applies directly when \(n\) is odd and the observations are ordered. For even \(n\), average the two central observations.
Mode — the most frequent value
The mode is the value that appears most often. A dataset may have one mode, multiple modes, or no mode if every value occurs equally often.
03. Measuring spread
Two datasets can have the same mean but very different distributions. Measures of spread help us understand how scattered the values are.
Range
Variance
Variance measures the average squared distance from the mean. The following is the population variance formula:
For a sample, the commonly used unbiased estimator of population variance divides the sum of squared deviations by \(n-1\):
Standard deviation
Standard deviation is the square root of variance, so it is expressed in the same units as the original data.
04. Quartiles and the interquartile range
Quartiles divide ordered data into four parts. The first quartile \(Q_1\) marks the lower quarter, while the third quartile \(Q_3\) marks the upper three-quarters.
The interquartile range describes the spread of the middle half of the observations. It is less affected by extreme values than the full range.
Different textbooks and software can use different quartile conventions, especially for small datasets. State the method when precision matters.
05. Distributions and outliers
A distribution describes how values are spread across a dataset. Histograms show frequencies across intervals; bar charts compare categories.
- Symmetric: the two sides have roughly similar shapes.
- Right-skewed: a longer tail extends towards larger values.
- Left-skewed: a longer tail extends towards smaller values.
- Outlier: a value unusually far from the rest of the data.
A common rule flags observations below \(Q_1-1.5\,\mathrm{IQR}\) or above \(Q_3+1.5\,\mathrm{IQR}\) as potential outliers. This is a useful screening rule, not proof that a value is an error.
06. Correlation does not prove causation
Correlation describes how two variables vary together. A positive relationship means larger values of one variable tend to accompany larger values of the other; a negative relationship means they tend to move in opposite directions.
A correlation near \(1\) indicates a strong positive linear relationship, while a value near \(-1\) indicates a strong negative linear relationship. A value near \(0\) means little linear correlation, but a nonlinear relationship may still exist.
Think critically
If two things happen together, one does not necessarily cause the other. A third factor, coincidence, or the way data was collected may explain the relationship.
07. Interactive data explorer
Enter numbers separated by commas to calculate the mean, median, mode, range, and population standard deviation.
Median: 4
Mode: 4
Range: 7
Population standard deviation: 2.37
08. Practice
Try solving these questions before revealing the answers.