الرياضيات
The Interquartile Range
Q3 minus Q1, the spread of the middle half. Worked on bus waiting times, with the reason exam answers prefer it to the range when one value is extreme.
Interquartile range
The interquartile range is the upper quartile minus the lower quartile: the width of the middle half of an ordered data set.
- يُسمّى أيضاً
- IQR
- أين يقابله الطلاب
- Grade 8 alongside quartiles, and at GCSE and IGCSE wherever two box plots or cumulative frequency curves have to be compared in words.
الإجابة باختصار
The interquartile range is Q3 minus Q1, the spread of the middle half of an ordered data set. For bus waiting times of 2, 3, 4, 4, 5, 6, 7, 8, 9, 11 and 34 minutes, Q1 is 4 and Q3 is 9, so the interquartile range is 5 minutes.
مثال
IQR = Q3 − Q1 = 9 − 4 = 5 minutes, while the range = 34 − 2 = 32 minutes
Both numbers describe the same eleven waiting times, and they tell almost opposite stories. The range says the times vary by half an hour, which is true but is entirely the work of one bus that never came. The interquartile range says that on a typical day the wait lands somewhere in a five-minute window, which is the more useful thing to know if you are deciding when to leave the house.
It measures the middle half
Cutting a data set at Q1 and Q3 leaves a quarter of the values below, a quarter above, and half in between. The interquartile range is the width of that middle block, so a small value means the bulk of the data is tightly grouped and a large one means it is spread out — regardless of what the extremes are doing.
It is a single number describing spread, which is why it is never quoted on its own. An answer of "the IQR is 5 minutes" says nothing about whether the waits are long or short; pairing it with the median gives a centre and a spread, and that pair describes the data properly.
Why the range gets abandoned
The range depends on exactly two values — the smallest and the largest — and those are the two most likely to be unusual. One 34-minute wait, one mis-typed measurement, one student who was ill on test day, and the range changes completely while the data as a whole has barely moved.
The interquartile range throws away the top quarter and the bottom quarter before measuring, so a single extreme value on either end cannot touch it. That is the trade: you give up any information about the tails in exchange for a figure that stays steady. When a question asks which measure of spread is more appropriate and the data contains an obvious extreme, this is the answer it wants, and the reason is that the IQR is not affected by that value.
On a box plot, it is the box
A box plot draws the box from Q1 to Q3, so the width of the box is the interquartile range, and the whiskers reach out to the smallest and largest values, so the full width is the range. Reading both off the diagram takes seconds once you know which is which.
Comparison questions have a predictable shape, and the marks split evenly. One statement should compare centres using the medians, and one should compare spreads using the interquartile ranges — and both should be written in the context of the question rather than as bare numbers. "The second bus route has a smaller IQR, so its waiting times are more consistent" earns the mark; "the second IQR is smaller" usually does not.
أسئلة شائعة
Can the interquartile range be zero?
Yes, when at least half the data takes the same value. If 30 of 50 students all scored 7 out of 10, both quartiles can land on 7 and the IQR is 0, meaning the middle half of the data does not vary at all. It is unusual in continuous data and common in survey data with few options.
Is a larger interquartile range better or worse?
Neither on its own — it depends on what you want. For bus times, exam marks in a class you are teaching, or manufacturing measurements, a small IQR means consistency and is usually good. For something like the range of ability a school takes in, a large IQR is simply a fact about the intake, not a fault.
How is the IQR used to find outliers?
The standard rule flags anything more than 1.5 × IQR beyond either quartile. With Q1 = 4 and Q3 = 9 the IQR is 5, so 1.5 × 5 = 7.5 and the boundaries are −3.5 and 16.5. The 34-minute wait sits well past the upper boundary and is an outlier by that rule.
Do I need the IQR or the standard deviation?
For GCSE and IGCSE Mathematics, the interquartile range — standard deviation is not on those specifications. Use the IQR when the data has extreme values or when you are working from quartiles and box plots anyway; standard deviation belongs to courses that carry it, such as GCSE Statistics and A level.
آخر تحديث
معرفة الكلمة ليست كاستخدامها
يستطيع المعلّم أن يرى الطالب وهو يستعمله في سؤال، فيتبيّن أين يتوقف الفهم بالضبط. الحصة الأولى مجانية.