Chapter 13 - Statistics

Calculate the mean, variance, and standard deviation for the following continuous frequency distribution using the shortcut step-deviation method:
Classes: $30-40, 40-50, 50-60, 60-70, 70-80, 80-90, 90-100$;
Frequencies: $3, 7, 12, 15, 8, 3, 2$.

Find the mean deviation about the median for the grouped continuous data:
Classes: $0-10, 10-20, 20-30, 30-40, 40-50, 50-60$;
Frequencies: $6, 7, 15, 16, 4, 2$.

The mean and standard deviation of $100$ observations were calculated as $40$ and $5.1$, respectively, by a student who took by mistake $50$ instead of $40$ for one observation. What are the correct mean and standard deviation?

Two samples of sizes $n_1 = 50$ and $n_2 = 100$ have means $\bar{x}_1 = 54.1$ and $\bar{x}_2 = 50.3$ and standard deviations $\sigma_1 = 8$ and $\sigma_2 = 7$, respectively. Find the mean and standard deviation of the combined sample of size $150$.

Find the mean, variance, and standard deviation of the continuous frequency distribution:
Diameter (mm): $33-36, 37-40, 41-44, 45-48, 49-52$;
Number of screws: $15, 17, 21, 22, 25$.

If $\bar{x}$ is the mean and $\sigma^2$ is the variance of $n$ observations $x_1, x_2, \dots, x_n$, prove that the variance of $ax_1, ax_2, \dots, ax_n$ is $a^2\sigma^2$, and hence find the variance of $5, 10, 15, 20, 25$.

The frequencies of two series $X$ and $Y$ with the same mean $25$ are given. Series $X$ has standard deviation $\sigma_X = 4.2$ and Series $Y$ has $\sigma_Y = 5.6$. Which series is more dispersed? Justify algebraically using the coefficient of variation.

Calculate the mean deviation about the mean for the continuous grouped distribution:
Age (years): $10-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80$;
Number of persons: $4, 8, 10, 16, 12, 6, 4$.