Question

Use the frequency distribution shown below to construct an expanded frequency distribution high temperatures (F).

Class	$20-30$	$31-41$	$42-52$	$53-63$	$64-74$	$75-85$	$86-96$
Frequency	$17$	$41$	$67$	$68$	$79$	$67$	$26$

Answer 1

Step-1: Find the mid-point of the class interval:

The mid-point is given by:

$\frac{\text{upper limit}+\text{lower limit}}{2}$
The mid-point to each of class interval is:

1. $\frac{30+20}{2}=\frac{50}{2}=25$
2. $\frac{41+31}{2}=\frac{72}{2}=36$
3. $\frac{52+42}{2}=\frac{94}{2}=47$
4. $\frac{63+53}{2}=\frac{116}{2}=58$
5. $\frac{74+64}{2}=\frac{138}{2}=69$
6. $\frac{85+75}{2}=\frac{160}{2}=80$
7. $\frac{96+86}{2}=\frac{182}{2}=91$

Step-2: Find Relative frequency:

The relative frequency gives the frequency in terms of percentage is given by$\frac{\text{Frequency}}{\text{Total Frequency}}\times 100$.

The total of the frequency is $17+41+67+68+79+67+26=365$.

The Relative frequency corresponding to each of class interval is:

1. $\frac{17}{365}\times 100=4.66\%$
2. $\frac{41}{365}\times 100=11.23\%$
3. $\frac{67}{365}\times 100=18.36\%$
4. $\frac{68}{365}\times 100=18.63\%$
5. $\frac{79}{365}\times 100=21.64\%$
6. $\frac{67}{365}\times 100=18.36\%$
7. $\frac{26}{365}\times 100=7.12\%$

Step-3: Find Cumulative frequency:

The cumulative frequency is given by a frequency added with all back class frequencies starting first as same.

The Cumulative frequency corresponding to each of class interval is:

1. $17$
2. $17+41=58$
3. $17+41+67=125$
4. $17+41+67+68=193$
5. $17+41+67+68+79=272$
6. $17+41+67+68+79+67=339$
7. $17+41+67+68+79+67+26=365$

Hence the combined result is as follow:

Class	Frequency	Mid-Point	Relative frequency	Cumulative frequency
$20-30$	$17$	$25$	$4.66\%$	$17$
$31-41$	$41$	$36$	$11.23\%$	$58$
$42-52$	$67$	$47$	$18.36\%$	$125$
$53-63$	$68$	$58$	$18.63\%$	$193$
$64-74$	$79$	$69$	$21.64\%$	$272$
$75-85$	$67$	$80$	$18.36\%$	$339$
$86-96$	$26$	$91$	$7.12\%$	$365$