1. Measures of spread: Calculating variance from a frequency table
The variance is given by either of the following two formulas which are equivalent:
𝜎 2 = 𝑘=1 𝑛 𝑥 𝑘 − 𝑥 2 𝑓 𝑘 𝑘=1 𝑛 𝑓 𝑘
𝜎 2 = 𝑘=1 𝑛 𝑥 𝑘 2 𝑓 𝑘 𝑘=1 𝑛 𝑓 𝑘 − 𝑥 2

2. Proof of 𝜎 2 = 𝑘=1 𝑛 𝑥 𝑘 − 𝑥 2 𝑓 𝑘 𝑘=1 𝑛 𝑓 𝑘 :
𝑘=1 𝑛 𝑥 𝑘 − 𝑥 2 𝑓 𝑘 𝑘=1 𝑛 𝑓 𝑘 = 𝑥 1 − 𝑥 2 𝑓 𝑥 2 − 𝑥 2 𝑓 2 +…+ 𝑥 𝑛 − 𝑥 2 𝑓 𝑛 𝑓 1 + 𝑓 2 +…+ 𝑓 𝑛
= 𝑥 1 2 −2 𝑥 1 𝑥 − 𝑥 2 𝑓 𝑥 2 2 −2 𝑥 2 𝑥 − 𝑥 2 𝑓 2 +…+ 𝑥 𝑛 2 −2 𝑥 𝑛 𝑥 − 𝑥 2 𝑓 𝑛 𝑓 1 + 𝑓 2 +…+ 𝑓 𝑛
= 𝑥 𝑓 1 + 𝑥 𝑓 2 +…+ 𝑥 𝑛 2 𝑓 𝑛 −2 𝑥 𝑥 1 𝑓 1 + 𝑥 2 𝑓 2 +…+ 𝑥 𝑛 𝑓 𝑛 + 𝑥 2 𝑓 1 + 𝑓 2 +…+ 𝑓 𝑛
The red part is equal to −2 𝑥 2 , giving
= 𝑥 𝑓 1 + 𝑥 𝑓 2 +…+ 𝑥 𝑛 2 𝑓 𝑛 𝑓 1 + 𝑓 2 +…+ 𝑓 𝑛 − 𝑥 as expected.

3. Now do Exercise 3C. 
After completing the last section called Choosing how to represent data, in chapter 3, do Miscellaneous exercise 3.