الأسبوع 9 · الوحدة ١٠ «تحليل البيانات»: أنواع البيانات، المتوسط والوسيط والمنوال، الرباعيات، التباين والانحراف المعياري، المخطط المبعثر والارتباط، الانحدار · 3 دروس
2
Spread: quartiles, variance and standard deviation
ICT Secondary 1 Term 1, Chapter 10: pp. 123–141
GoalFind the quartiles, the variance and the standard deviation of a small list and compare two spreads.
New words
quartile · one of the three values that cut a sorted list into four partsbox plot · a picture of the minimum, Q1, median, Q3 and maximumvariance · the average of the squared distances from the meanstandard deviation · the square root of the variance
Explanation
Two lists can have the same mean and still be very different, so we also measure the spread. Quartiles cut a sorted list into four parts. Q2 is the median. Q1 is the median of the lower half and Q3 the median of the upper half; in this lesson, when the number of values is odd, the median itself is left out of both halves. For 2, 4, 4, 4, 5, 5, 7, 9 (eight values): the median is (4 + 5) ÷ 2 = 4.5, Q1 = (4 + 4) ÷ 2 = 4 and Q3 = (5 + 7) ÷ 2 = 6. The five numbers minimum 2, Q1 4, median 4.5, Q3 6, maximum 9 are what a box plot shows. Variance uses every value: find the mean (5), the distance of each value from it, square each distance, and take the average. Here the squared distances are 9, 1, 1, 1, 0, 0, 4, 16, which add to 32, and 32 ÷ 8 = 4. We use the population formula in this lesson, dividing by n, the number of values. The standard deviation is the square root of the variance: √4 = 2. It has the same unit as the data. A second list 3, 5, 5, 5, 5, 5, 5, 7 also has mean 5, but its variance is (4 + 4) ÷ 8 = 1 and its standard deviation is 1: the values are closer to the mean.
Value
2
4
4
4
5
5
7
9
distance from mean 5
−3
−1
−1
−1
0
0
2
4
squared distance
9
1
1
1
0
0
4
16
Worked examples
Minimum
Q1
Median
Q3
Maximum
2
4
4.5
6
9
The five-number summary of 2, 4, 4, 4, 5, 5, 7, 9 for a box plot.
List A
List B
mean
5
5
variance
4
1
standard deviation
2
1
Same mean, different spread. List A is 2, 4, 4, 4, 5, 5, 7, 9; list B is 3, 5, 5, 5, 5, 5, 5, 7. Both have mean 5. A has standard deviation 2; B has 1. So B is more consistent.
Part
Values
Median of the part
lower half
1, 3, 4
3 (Q1)
middle
6
6 (Q2)
upper half
8, 9, 12
9 (Q3)
Odd number of values: 1, 3, 4, 6, 8, 9, 12. The median is 6 (the fourth value). Leaving it out, the lower half is 1, 3, 4 with Q1 = 3, and the upper half is 8, 9, 12 with Q3 = 9.
Class activity
For the list 4, 4, 8, 8 find the mean, the squared distances from the mean, the variance (divide by 4) and the standard deviation. Then invent another list of four numbers with the same mean and a smaller standard deviation.
Practice
1
Find Q1, median and Q3 of 2, 3, 5, 7, 8, 9, 11, 12.
