Let’s solve this problem step by step:

Given Data:

19, 23, 25, 28, 30, 32, 33, 33

1. Mean:

The mean is calculated as: $\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$ $\text{Sum} = 19 + 23 + 25 + 28 + 30 + 32 + 33 + 33 = 243$ $\text{Mean} = \frac{243}{8} = 30.4$

2. Median:

The median is the middle value(s) when the data is ordered. Since there are 8 values (an even number): $\text{Median} = \frac{\text{4th value + 5th value}}{2}$ $\text{Median} = \frac{28 + 30}{2} = 29.0$

3. Mode:

The mode is the most frequently occurring value(s). In the dataset:

• 33 appears twice, while all other values appear once. $\text{Mode} = 33$

4. Population Standard Deviation:

The formula for population standard deviation is: $\sigma = \sqrt{\frac{\sum{(x_i - \mu)^2}}{N}}$ Where:

• $x_i$ = each data value,
• $\mu$ = mean,
• $N$ = number of values.

Step 1: Calculate deviations from the mean ($x_i - \mu$): $\mu = 30.4$