Discrete probability distribution - sample mean and variance

Below is a frequency table for lengths in cm of a species of tropical fish

(a) Calculate the sample mean.
(b) Find the sample variance.

Use the definition to calculate the sample mean

\[ \begin{align} \text{mean} &= \sum_{i=1}^n x_i \times \frac{f_i}{43} \\ &= \frac{10 \times 4+11 \times 10+12 \times 14+13 \times 9 +14 \times 6}{43} \\ &= \frac{519}{43} \\ &= 12.07 \end{align}\ \]

Use the definition to calculate the sample variance. Set mean = m

\[ \begin{align} s^2 &= \frac{\sum_{i=1}^n (x_i - \text{m})^2 \times x_i}{N-1} \\ &= \frac{4(10 - 12.07)^2 + 10(11 - 12.07)^2 + 14(12 - 12.07)^2}{42} \\ &+ \frac{9(13 - 12.07)^2 + 6(14 - 12.07)^2}{42} \\ &= \frac{58.79}{42} \\ &= 1.40 \end{align}\ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

Above is a frequency table for lengths in cm of a species of tropical fish

(a) Calculate the sample mean.
(b) Find the sample variance.