Statistics - Interpreting Statistical Data

Calculate the estimated mean for the following grouped data: Class $0 < x \le 10$: Frequency $4$ Class $10 < x \le 20$: Frequency $6$

1. Find the midpoints ($x$) for each class: Midpoint of $(0, 10)$ is $5$. Midpoint of $(10, 20)$ is $15$.
2. Multiply midpoints by frequencies ($f \times x$): $4 \times 5 = 20$ $6 \times 15 = 90$
3. Sum the frequencies and products: $\sum f = 4 + 6 = 10$ $\sum f x = 20 + 90 = 110$
4. Calculate Mean: $\bar{x} = \frac{110}{10} = 11$

For grouped data, we assume all values in a class are represented by the midpoint of that class interval.

A histogram has a bar for the class $20 < x \le 50$. The frequency of this class is $150$. Calculate the frequency density for this bar.

$Class \ Width = 50 - 20 = 30$ $Frequency = 150$ $Frequency \ Density = \frac{150}{30}$ $Frequency \ Density = 5$

In a histogram with unequal class widths, the height of the bar is the frequency density, such that $Area = Width \times Height = Frequency$.

On a cumulative frequency curve representing $200$ students, find the positions of the Median and the Interquartile Range.

1. Median Position: $0.50 \times 200 = 100^{th} \text{ value}$
2. Lower Quartile ($Q_1$) Position: $0.25 \times 200 = 50^{th} \text{ value}$
3. Upper Quartile ($Q_3$) Position: $0.75 \times 200 = 150^{th} \text{ value}$
4. $IQR$ is the horizontal distance between the $x$-values at the $150^{th}$ and $50^{th}$ positions: $IQR = Q_3 - Q_1$

Cumulative frequency graphs allow us to estimate percentiles. The median is the $50^{th}$ percentile, $Q_1$ is the $25^{th}$, and $Q_3$ is the $75^{th}$.