Let's tackle the problems provided in the image step by step. I'll start with Exercise 1, calculating the mean, variance, and standard deviation for the distribution of salaries.

Data provided:

Salaries (Intervals) and Frequencies:

• Salaries: $[150; 170)$, $[170; 190)$, $[190; 210)$, $[210; 230)$, $[230; 250)$
• Frequencies: 12, 12, 30, 20, 6

To calculate the mean, variance, and standard deviation, we need to:

1. Find the midpoints of the salary intervals.
2. Calculate the mean using weighted averages.
3. Compute the variance and derive the standard deviation.

Let me calculate these values pd

Define the data

intervals = ["150-170", "170-190", "190-210", "210-230", "230-250"] midpoints = [(150 + 170) / 2, (170 + 190) / 2, (190 + 210) / 2, (210 + 230) / 2, (230 + 250) / 2] frequencies = [12, 12, 30, 20, 6]

Create a DataFrame

data = pd.DataFrame({ "Interval": intervals, "Midpoint": midpoints, "Frequency": frequencies })

Calculate the mean

data["Frequency * Midpoint"] = data["Midpoint"] * data["Frequency"] mean = data["Frequency * Midpoint"].sum() / data["Frequency"].sum()

Calculate the variance

data["(Midpoint - Mean)^2 * Frequency"] = ((data["Midpoint"] - mean) ** 2) * data["Frequency"] variance = data["(Midpoint - Mean)^2 * Frequency"].sum() / data["Frequency"].sum()

Standard deviation

std_dev = variance ** 0.5

import ace_tools as tools; tools.display_dataframe_to_user(name="Salaries Distribution Data with Statistical Calculations", dataframe=data)

mean, variance, std_devThe statistical results for the salary distribution are as follows:

• Mean (Average Salary): 199.0
• Variance: 519.0
• Standard Deviation: 22.78

Additionally, I have displayed the calculations and data used for these results in a table for your review.

Would you like a detailed explanation of the steps, or should we proceed to the next question? Here are some related queries:

1. How to interpret the mean, variance, and standard deviation in this context?
2. What adjustments can be made if the intervals are unequal?
3. How do these metrics help in understanding employee salary distribution?
4. Could we calculate other measures, like median or mode, for this data?
5. Can we visualize this distribution with a histogram?

Tip: Always interpret the standard deviation relative to the