Question

fire calls the frequency distribution shows the number of fire calls per month a volunteer fire department received over a two - year period. find the variance and standard deviation for the data. round your answers to at least one decimal place.
class frequency
3 - 5 2
6 - 8 4
9 - 11 11
12 - 14 5
15 - 17 2
variance=
standard deviation=
send data to excel

Explanation:

Step1: Calculate the mid - point of each class

For class \(3 - 5\), mid - point \(x_1=\frac{3 + 5}{2}=4\)
For class \(6 - 8\), mid - point \(x_2=\frac{6+8}{2}=7\)
For class \(9 - 11\), mid - point \(x_3=\frac{9 + 11}{2}=10\)
For class \(12 - 14\), mid - point \(x_4=\frac{12+14}{2}=13\)
For class \(15 - 17\), mid - point \(x_5=\frac{15 + 17}{2}=16\)

Step2: Calculate the mean \(\bar{x}\)

The formula for the mean of a frequency distribution is \(\bar{x}=\frac{\sum_{i = 1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\)
Here, \(f_1 = 2\), \(f_2=4\), \(f_3 = 11\), \(f_4=5\), \(f_5=2\)
\(\sum_{i=1}^{n}f_ix_i=2\times4 + 4\times7+11\times10+5\times13+2\times16\)
\(=8+28 + 110+65+32\)
\(=243\)
\(\sum_{i=1}^{n}f_i=2 + 4+11+5+2=24\)
\(\bar{x}=\frac{243}{24}=10.125\)

Step3: Calculate the variance \(s^{2}\)

The formula for the variance of a frequency distribution is \(s^{2}=\frac{\sum_{i = 1}^{n}f_i(x_i-\bar{x})^{2}}{\sum_{i=1}^{n}f_i-1}\)
\((x_1-\bar{x})^{2}=(4 - 10.125)^{2}=(- 6.125)^{2}=37.515625\)
\((x_2-\bar{x})^{2}=(7 - 10.125)^{2}=(-3.125)^{2}=9.765625\)
\((x_3-\bar{x})^{2}=(10 - 10.125)^{2}=(-0.125)^{2}=0.015625\)
\((x_4-\bar{x})^{2}=(13 - 10.125)^{2}=(2.875)^{2}=8.265625\)
\((x_5-\bar{x})^{2}=(16 - 10.125)^{2}=(5.875)^{2}=34.515625\)
\(\sum_{i = 1}^{n}f_i(x_i-\bar{x})^{2}=2\times37.515625+4\times9.765625+11\times0.015625+5\times8.265625+2\times34.515625\)
\(=75.03125+39.0625+0.171875+41.328125+69.03125\)
\(=224.625\)
\(s^{2}=\frac{224.625}{24 - 1}=\frac{224.625}{23}\approx9.8\)

Step4: Calculate the standard deviation \(s\)

The formula for the standard deviation is \(s=\sqrt{s^{2}}\)
\(s=\sqrt{9.8}\approx3.1\)

Answer:

Variance \( = 9.8\), Standard deviation \(=3.1\)