Q.Consider the temperature (in Celsius) of 7 days of a week as 34, 34, 27, 28, 27, 34, 34. Identify the appropriate statistical technique to be used to calculate the following:

This is a plain theory question asking you to identify the statistical technique for three measures and then compute them from the given temperature data.

Given data:

Temperatures (°C) over 7 days: $34, 34, 27, 28, 27, 34, 34$.

(a) Find the average temperature.

The appropriate technique is the arithmetic mean.

The arithmetic mean is calculated as:

where $x_i$ are the individual observations and $n$ is the number of observations.

For the given data:

The average temperature of the week is 31.14 °C.

(b) Find the temperature Range of that week.

The appropriate technique is the range, which measures the spread of data.

Range is calculated as:

From the data:

• Maximum temperature = $34$ °C
• Minimum temperature = $27$ °C

The temperature range of the week is 7 °C.

(c) Find the standard deviation temperature.

The appropriate technique is the standard deviation, which measures the dispersion of data around the mean.

Standard deviation is calculated as:

where $\bar{x}$ is the mean.

Using $\bar{x} = 31.14$:

| $x_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ | …

Understanding the Statistical Techniques

When we have a dataset of numerical observations—here, daily temperatures—we need different measures to summarize and understand its characteristics. Each part of this question asks for a specific statistical property, and each requires its own technique.

The temperatures for the week are: 34, 34, 27, 28, 27, 34, 34 (in °C).

(a) Average Temperature – Arithmetic Mean

The arithmetic mean is the appropriate technique to find the average. It gives us the central tendency by summing all values and dividing by the count.

Why the mean? When we want a single representative value that balances all observations equally, the mean is the standard choice. It tells us what a "typical" temperature might be if the heat were distributed evenly across the week.

Formula:

where $x_i$ are the individual temperatures and $n$ is the number of days.

import numpy as np

temperatures = [34, 34, 27, 28, 27, 34, 34]

# Calculate mean
mean_temp = np.mean(temperatures)
# Or manually: sum(temperatures) / len(temperatures)

print(f"Average temperature: {mean_temp}°C")

Calculation:

Output:

Average temperature: 31.14°C

Rounded to two decimal places: 31.14°C

(b) Temperature Range

The range is the appropriate technique to measure the spread between the highest and lowest temperatures.

Why the range? It gives us the simplest measure of variability—how much the temperature fluctuated during the week. A large range means high variability; a small range means stable conditions.

Formula:

# Calculate range
max_temp = max(temperatures)
min_temp = min(temperatures)
temp_range = max_temp - min_temp

print(f"Maximum temperature: {max_temp}°C")
print(f"Minimum temperature: {min_temp}°C")
print(f"Temperature range: {temp_range}°C")

Calculation:

Output:

Maximum temperature: 34°C
Minimum temperature: 27°C
Temperature range: 7°C

(c) Standard Deviation

The standard deviation is the appropriate technique to measure how much individual temperatures deviate from the mean.

Why standard deviation? Unlike the range, which only uses two values, standard deviation considers every observation. It tells us how tightly clustered the temperatures are around the average. A small standard deviation means temperatures were consistent; a large one means they varied significantly.

Formula (Population standard deviation):

where $\mu$ is the mean and $n$ is the number of observations.

# Calculate standard deviation (population)
std_dev = np.std(temperatures)  # Population std by default in NumPy

print(f"Standard deviation: {std_dev}°C")

# Step-by-step calculation for clarity …