To find the mean (denoted as [tex]\(\bar{X}\)[/tex]) in a continuous series given [tex]\(\Sigma f = 20p + 7\)[/tex] and [tex]\(\Sigma fm = 100p + 35\)[/tex], we follow these steps:
1. Write down the formulas provided:
- Total frequency: [tex]\(\Sigma f = 20p + 7\)[/tex]
- Sum of all midpoints (frequency times midpoint): [tex]\(\Sigma fm = 100p + 35\)[/tex]
2. Recall the formula for the mean in a continuous series:
- [tex]\(\bar{X} = \frac{\Sigma fm}{\Sigma f}\)[/tex]
3. Substitute [tex]\(\Sigma f\)[/tex] and [tex]\(\Sigma fm\)[/tex] into the formula for the mean:
[tex]\[\bar{X} = \frac{100p + 35}{20p + 7}\][/tex]
4. Simplify the expression (if possible):
- There is no common factor for [tex]\(100p+35\)[/tex] and [tex]\(20p+7\)[/tex], so the expression remains as is.
Thus, the mean [tex]\(\bar{X}\)[/tex] is:
[tex]\[
\bar{X} = \frac{100p + 35}{20p + 7}
\][/tex]
So, in conclusion, the mean of the continuous series is:
[tex]\[
\boxed{\frac{100p + 35}{20p + 7}}
\][/tex]
