☰ خطة العام · الإحصاء (بالإنجليزية)

Box plots and the semi-interquartile range

الأسبوع 9 · التمثيل الصندوقي ونصف المدى الربيعي: الأعداد الخمسة ورسم الصندوق وقراءته، و⁦(Q₃ − Q₁) ÷ 2⁩ من القائمة والساق والأوراق والجدول التكراري · 3 دروس
3

The semi-interquartile range Q = (Q3 − Q1) ÷ 2 from a list, a stem-and-leaf plot and a frequency table, and comparing spread

Statistics Sec 3 Term 1, Unit 2, Lessons 2–3
GoalFind the semi-interquartile range Q = (Q₃ − Q₁) ÷ 2 from a list, a stem-and-leaf plot and a frequency table, and use it to compare spread.
New words
semi-interquartile range Q · half of the length of the box: (Q₃ − Q₁) ÷ 2interquartile range · Q₃ − Q₁, the length of the boxmeasure of spread · a number that tells how spread out the data are, such as the range or Q
Explanation

The range uses only the two extreme values, so one unusual value can change it a great deal. The semi-interquartile range ignores the extremes: it is half the length of the box, Q = (Q₃ − Q₁) ÷ 2, and it measures the spread of the middle half of the data. The whole difference Q₃ − Q₁ is called the interquartile range. To find Q, find Q₁ and Q₃ by the rule that fits the data (list or plot: the places (n + 1) ÷ 4 and 3(n + 1) ÷ 4; table: the places n ÷ 4 and 3n ÷ 4 and the formula), subtract and halve. The advantages of Q: extreme values hardly move it, and it is easy to calculate. Its disadvantage: it does not use all the values, because it ignores the lowest and the highest quarter. A small Q means that the middle of the data is close together; a large Q that it is spread out.

30354045505560
Worked examples
From a list. Twelve invented marks: 38, 45, 41, 52, 36, 47, 44, 50, 39, 42, 48, 55. Ordered: 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 52, 55. n + 1 = 13: places 3.25 and 9.75. Q₁ = 39 + 0.25 × 2 = 39.5 and Q₃ = 48 + 0.75 × 2 = 49.5. So Q = (49.5 − 39.5) ÷ 2 = 5. The range is 55 − 36 = 19.
An extreme value. Add one more mark, 120, to the same data (n = 13). Now Q₁ = (39 + 41) ÷ 2 = 40 and Q₃ = (50 + 52) ÷ 2 = 51, so Q = (51 − 40) ÷ 2 = 5.5, only a little more than before. But the range jumps from 19 to 120 − 36 = 84. Q resists the extreme value; the range does not.
ClassFrequencyLess thanCumulative frequency
00
0–4104
10–132017
20–193036
30–104046
40–5045050
From a frequency table. Fifty pupils’ walking distances to school in a week: classes 0–, 10–, 20–, 30–, 40–50 with frequencies 4, 13, 19, 10, 4. Cumulative 0, 4, 17, 36, 46, 50; the orders are 12.5 and 37.5. Q₁ = 10 + (12.5 − 4) ÷ 13 × 10 = 16.54 and Q₃ = 30 + (37.5 − 36) ÷ 10 × 10 = 31.5. So Q = (31.5 − 16.54) ÷ 2 = 7.48 km.
Class activity

Two classes took the same quiz. Class X has Q = 3 and class Y has Q = 8, and the medians are equal. Write two sentences about which class is more consistent, and one about what Q cannot tell you.

Practice
1
The five numbers of a data set are 12, 20, 26, 34, 40. Find Q.
2
Find Q for the plot with rows 5 | 5 8 9, 6 | 2 4 6 8 9, 7 | 0 1 2 3 4 7 and 8 | 0 (key 5|5 = 55).
key 5|5 = 55
StemLeaves
55 8 9
62 4 6 8 9
70 1 2 3 4 7
80
3
Give the interquartile range and Q when Q₁ = 18.5 and Q₃ = 27.5.
4
Why does one very large value hardly change Q but change the range a lot?
5
Class X has median 62 and Q = 5; class Y has median 58 and Q = 9. Compare the classes.