To find the mean of a frequency distribution with intervals, we first need to find the midpoint of each interval.
We do that by adding the limits of each interval and dividing by 2:
1
[tex]\frac{40+44}{2}=42[/tex]2
[tex]\frac{45+49}{2}=47[/tex]3
[tex]\frac{50+54}{2}=52[/tex]4
[tex]\frac{55+59}{2}=57[/tex]5
[tex]\frac{60+64}{2}=62[/tex]Now, we multiply these midpoints by the frequency of each interval and sum them:
[tex]\begin{gathered} 3\cdot42+7\cdot47+11\cdot52+5\cdot57+2\cdot62 \\ 126+329+572+285+124=1436 \end{gathered}[/tex]Now, we divide this sum by the sum of the frequencies, and this will be the mean.
[tex]m=\frac{1436}{3+7+11+5+2}=\frac{1436}{28}=51.3[/tex]So, the mean calculated by using the frequency distribution is 51.3, which is lower than the actual mean of the data, 56.3.