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Standardization is a key concept in statistical analysis that allows us to compare different data sets or distributions by bringing them to a common scale. This is particularly useful when dealing with normal distributions.
In our example, standardization involves translating the mean height of women into the men's distribution framework, which has different mean and standard deviation.
You use the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

where:

• \( X \) is the value you wish to standardize (here, 64.1 inches),
• \( \mu \) is the mean of the distribution you are interested in (69.4 inches for men),
• \( \sigma \) is the distribution's standard deviation (3.1 inches for men).

Through standardization, we convert raw measurements into Z-scores, allowing for easier comparison across diverse datasets.

The Z-score measures how many standard deviations a data point is from the mean.
In our context, it indicates how the average height of women compares to the men's height distribution.
By calculating the Z-score, you determine that the mean women's height (64.1 inches) is about -1.71 standard deviations below men's average height (69.4 inches).
The formula used is

• \( Z = \frac{64.1 - 69.4}{3.1} \)
• \( Z \approx -1.71 \).

This negative Z-score means the women's average height is significantly shorter than the men's.
Z-scores make it possible to identify and interpret the position of a single data point relative to the entire distribution.

Probability helps us quantify the chance of an event occurring. In this exercise, we're interested in knowing the probability that a randomly chosen man is taller than the average height of women.
Once the Z-score is calculated, it becomes straightforward to find this probability using Z-tables or statistical calculators.
The Z-table provides cumulative probabilities, meaning it expresses the likelihood of the height being below a certain Z-score.
For a Z-score of -1.71, the probability is approximately 96.57% that a man would be shorter than 64.1 inches.
Subtracting this from 100%, the probability that a man is taller is 3.43%. Thus, probability helps predict outcomes based on established statistical models.

Statistical analysis is the backbone of understanding data.
It involves collecting, reviewing, and interpreting data to uncover patterns or trends.
Analyzing the height distributions of men and women involves steps like determining the mean, standard deviation, and Z-scores.
Each component offers insight into the population characteristics.
In this example, statistical analysis helps answer the practical question of height comparison using the normal distribution as the framework.
This enables us to deduce conclusions like how many men are statistically expected to exceed a certain height, enhancing our comprehension of data trends.