Measures of Spread (AI SL)
Two data sets can share the same mean and still behave completely differently - measures of spread are how you quantify that difference. This page covers the range, the interquartile range, and the variance and standard deviation you'll pull straight off your GDC, with worked examples and the mistakes that lose marks. It's part of the broader Descriptive Statistics topic.
15 questions on this sub-topic.
Range and IQR
Covered under IB syllabus reference SL4.3: measures of dispersion (IQR, standard deviation and variance), calculated using technology, plus the effect of constant changes such as adding or scaling every value.
Range
\(\text{Range} = \text{max} - \text{min}\)
Not in the formula booklet - prior knowledge. The total spread of the data, but it's very sensitive to a single extreme value.
Interquartile range
\(\text{IQR} = Q_3 - Q_1\)
Not in the formula booklet - prior knowledge. The spread of the middle 50% of the data, so extreme values don't distort it the way they distort the range.
Standard deviation and variance are always found using your GDC's 1-Variable Statistics, not by hand. Need the full syllabus wording? See Descriptive Statistics.
Worked examples
A data set has standard deviation \(6.\) State the variance.
Worked solution
Variance \(=6^2=36.\) A1
A data set is given below.
| 12 | 15 | 15 | 18 | 20 | 22 |
(a) Find the mean.
(b) Find the standard deviation (3 significant figures).
Worked solution
(a) Mean. Enter all six values in the GDC: \(\bar x=17.\) A1
(b) Standard deviation. Select \(\sigma x\) (not \(Sx\)): \(\sigma x\approx 3.37.\) A1
From a cumulative frequency graph of 160 values, the 40th value is 18 and the 120th value is 31.
(a) State which quartiles these represent.
(b) Find the interquartile range.
Worked solution
(a) Which quartiles. \(\tfrac14\times160=40\) and \(\tfrac34\times160=120,\) M1
the 40th value is \(Q_1=18\) A1
the 120th value is \(Q_3=31.\) A1
(b) IQR. \(Q_3-Q_1=31-18\) M1
\(=13.\) A1
Common mistakes
- Reading \(Sx\) instead of \(\sigma x\). A GDC's 1-Var Stats screen shows both a sample standard deviation \((Sx)\) and a population standard deviation \((\sigma x)\) - IB questions almost always want \(\sigma x\).
- Quoting the range as a single extreme-sensitive summary. One unusually large or small value inflates the range even though the rest of the data is tightly packed - the IQR is the more robust measure when a question asks you to compare consistency.
- Forgetting how scaling affects spread. Adding a constant to every value shifts the mean but leaves the standard deviation unchanged; multiplying every value by a constant \(k\) multiplies the standard deviation by \(|k|\), not by \(k\) alone.
Ready to practise properly?
15 measures-of-spread questions, marked instantly like the real exam.
Quick answers
What is the difference between range and IQR?
The range is \(\text{max}-\text{min}\) and uses every value, including any extremes. The IQR is \(Q_3-Q_1\), the spread of the middle 50% of the data, so it isn't distorted by an unusually high or low value.
Should I read sigma-x or Sx on my GDC?
Read \(\sigma x\) (population standard deviation), not \(Sx\) (sample standard deviation), unless a question specifically asks for a sample statistic - this is the single most common lost mark in this topic. See using your GDC on the full topic page for the full key sequence.