To analyze the given dataset and calculate the requested values (arithmetic mean, range, standard deviation, median, and mode), I'll proceed with the following steps based on the data provided:

Dataset:

The numbers are:
6, 6, 8, 10, 12, 20, 22, 21, 22, 19, 17, 15, 17, 19, 8, 22.

Calculations:

1. Arithmetic Mean ($\bar{x}$):

The formula for the arithmetic mean is:
$\bar{x} = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

2. Range:

The range is calculated as:
$\text{Range} = \text{Maximum value} - \text{Minimum value}$

3. Standard Deviation ($s$):

The standard deviation formula is:
$s = \sqrt{\frac{1}{N} \sum (x_i - \bar{x})^2}$

4. Median:

To find the median:

• Sort the data.
• Find the middle value(s). If the number of data points is even, the median is the average of the two middle values.

5. Mode:

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

Let me extract and calculate these details step-by-step. Dataset provided data = [6, 6, 8, 10, 12, 20, 22, 21, 22, 19, 17, 15, 17, 19, 8, 22]

Step 1: Calculate Arithmetic Mean

n = len(data) mean = sum(data) / n

Step 2: Calculate Range

range_val = max(data) - min(data)

Step 3: Calculate Standard Deviation

squared_diffs = [(x - mean) ** 2 for x in data] std_dev = (sum(squared_diffs) / n) ** 0.5

Step 4: Calculate Median

sorted_data = sorted(data) if n % 2 == 0: # Even number of elements median = (sorted_data[n // 2 - 1] + sorted_data[n // 2]) / 2 else: # Odd number of elements median = sorted_data[n // 2]

Step 5: Calculate Mode

from collections import Counter freq_counts = Counter(data) max_freq = max(freq_counts.values()) modes = [key for key, val in freq_counts.items() if val == max_freq]

mean, range_val, std_dev, median, modes### Results:

1. Arithmetic Mean ($\bar{x}$): $15.25$
2. Range: $16$ (Maximum: 22, Minimum: 6)
3. Standard Deviation ($s$): $5.81$ (approx.)
4. Median: $17.0$
5. Mode: $22$ (occurs 3 times)

Would you like a detailed explanation of each calculation or further assistance with this problem?

Related Questions:

1. How is the standard deviation formula derived?
2. What happens to the mean if one extreme value (outlier) is added to the dataset?
3. Can a dataset have more than one mode? Why or why not?
4. What are the implications of a high standard deviation in a dataset?
5. How does the median differ from the mean in terms of robustness to outliers?

Tip:

To better understand the spread of data, compare the range and standard deviation. While the range shows the extremes, the standard deviation indicates how data points are distributed around the mean.