Let's walk through the steps to find the mean, standard deviation, and the z-score for the value 33.8 in this data set.

Step 1: Calculate the Mean (M)

To calculate the mean (M), we sum all the data points and divide by the number of data points.

The data set has the following values:

53, 33.8, 34.7, 46.6, 33.8, 49.2, 55.7, 56.8, 44.6, 29.3, 46, 46.6, 59, 33.2, 46.8, 51.5, 52.3, 40.1, 51.7, 44.6, 56.8, 36, 52.9, 47, 54.8, 46.8, 45.9, 37, 45.2, 30.6, 45.4, 61.1, 50.1, 55.4, 42.8, 47.9, 50.7, 40.8, 46.2, 41.9, 45, 43.2, 45.2, 49.4, 44.2, 49.2, 43.4, 45.2, 40.6, 59, 40.6, 33.8, 57.4, 59, 55.3, 41.7, 38.5, 51.7, 31, 46.6, 38.9, 40.3, 55.3, 45.8, 50.4, 32.6, 45.2, 51.5, 33.2, 28.6

First, we sum the values: $\text{Sum of data points} = 53 + 33.8 + 34.7 + 46.6 + 33.8 + 49.2 + 55.7 + 56.8 + 44.6 + 29.3 + \dots + 28.6$

Next, divide by the number of data points. The number of values in the data set is 63.

So, the mean (M) is: $M = \frac{\text{Sum of data points}}{63}$

Step 2: Calculate the Standard Deviation (SD)

The formula for standard deviation is: $S = \sqrt{\frac{\sum (x_i - M)^2}{n}}$ Where:

• $x_i$ are the individual data points
• $M$ is the mean
• $n$ is the number of data points (63)

We calculate the squared deviations from the mean for each data point, sum them up, and divide by the total number of data points. The square root of this gives us the standard deviation.

Step 3: Calculate the Z-Score

The z-score of a given value is calculated using the formula: $z = \frac{x - M}{S}$ Where:

• $x$ is the value (33.8 in this case)
• $M$ is the mean
• $S$ is the standard deviation

Let's go ahead and calculate the mean, standard deviation, and z-score. numpy as np

Data set

data = [ 53, 33.8, 34.7, 46.6, 33.8, 49.2, 55.7, 56.8, 44.6, 29.3, 46, 46.6, 59, 33.2, 46.8, 51.5, 52.3, 40.1, 51.7, 44.6, 56.8, 36, 52.9, 47, 54.8, 46.8, 45.9, 37, 45.2, 30.6, 45.4, 61.1, 50.1, 55.4, 42.8, 47.9, 50.7, 40.8, 46.2, 41.9, 45, 43.2, 45.2, 49.4, 44.2, 49.2, 43.4, 45.2, 40.6, 59, 40.6, 33.8, 57.4, 59, 55.3, 41.7, 38.5, 51.7, 31, 46.6, 38.9, 40.3, 55.3, 45.8, 50.4, 32.6, 45.2, 51.5, 33.2, 28.6 ]

Calculate mean

mean = np.mean(data)

Calculate standard deviation

std_dev = np.std(data, ddof=0)

Calculate z-score for the value 33.8

z_score = (33.8 - mean) / std_dev

mean, std_dev, z_scoreHere are the results based on the data:

1. Mean (M): 45.52 (rounded to 1 decimal place as instructed)
2. Standard Deviation (SD): 8.03 (rounded to 2 decimal places as instructed)
3. Z-score for the value 33.8: -1.460 (rounded to 3 decimal places)

If you need further details or have any other questions, feel free to ask!

Related Questions:

1. What is the effect of outliers on the mean and standard deviation?
2. How can you visualize the distribution of this data using a boxplot?
3. How can the normality of a data set be tested mathematically?
4. What is the relationship between standard deviation and variance?
5. How would you interpret a z-score of -1.460 in practical terms?

Tip:

When calculating the standard deviation, it's important to remember that using the sample standard deviation formula (with $ddof=1$) provides a more accurate estimate for a sample of a population.