Exercise Facility Usage Analysis Finding Values 1.5 Standard Deviations From The Mean

by THE IDEN 86 views

In this statistical analysis, we delve into the exercise habits of clients at an exercise facility. We have a sample of 100 clients, and our focus is on understanding their usage patterns. Specifically, we're interested in X, which represents the number of days per week a randomly selected client uses the facility. The provided frequency distribution gives us a snapshot of these usage patterns. Our goal is to analyze this data, calculate key statistical measures, and interpret the findings to gain insights into client behavior. Understanding these patterns can help the facility optimize its services, tailor programs, and improve client engagement. This analysis will involve calculating the mean and standard deviation of the data, which are crucial for understanding the central tendency and variability of exercise facility usage among clients. We will then use these measures to determine specific data points, such as the number of days that falls a certain number of standard deviations away from the mean. By exploring these statistical concepts, we aim to provide a comprehensive understanding of client exercise habits and their implications for the facility.

The data we have is presented as a frequency distribution, which is a table that shows how many clients (frequency) use the facility a certain number of days per week (X). Let's break down the provided frequency distribution:

  • X = 0 days: 2 clients
  • X = 1 day: 13 clients
  • X = 2 days: 30 clients
  • X = 3 days: 29 clients
  • X = 4 days: 11 clients
  • X = 5 days: 8 clients
  • X = 6 days: 7 clients

Looking at this data, we can make some initial observations. It seems like most clients use the facility 2 or 3 days a week, as these frequencies are the highest. There are relatively few clients who use the facility 0, 5, or 6 days a week. This distribution gives us a general sense of the central tendency of usage, but to understand the data more precisely, we need to calculate statistical measures like the mean and standard deviation. The mean will tell us the average number of days clients use the facility, while the standard deviation will tell us how spread out the data is around the mean. A higher standard deviation would indicate more variability in usage patterns, while a lower standard deviation would suggest that clients tend to use the facility a similar number of days per week. These initial observations are crucial for setting the stage for a more in-depth statistical analysis.

The mean, often referred to as the average, is a fundamental measure of central tendency. It tells us the typical value in a dataset. To calculate the mean number of days clients use the exercise facility per week, we'll use the following formula for a frequency distribution:

Mean (μ) = Σ(X * f) / Σf

Where:

  • X is the number of days per week
  • f is the frequency (number of clients) for each X
  • Σ means "sum of"

Let's apply this formula to our data:

  1. Multiply each X value by its frequency (X * f):
    • 0 days: 0 * 2 = 0
    • 1 day: 1 * 13 = 13
    • 2 days: 2 * 30 = 60
    • 3 days: 3 * 29 = 87
    • 4 days: 4 * 11 = 44
    • 5 days: 5 * 8 = 40
    • 6 days: 6 * 7 = 42
  2. Sum these products (Σ(X * f)): 0 + 13 + 60 + 87 + 44 + 40 + 42 = 286
  3. Sum the frequencies (Σf): 2 + 13 + 30 + 29 + 11 + 8 + 7 = 100 (This is our sample size)
  4. Divide the sum of the products by the sum of the frequencies: 286 / 100 = 2.86

Therefore, the mean number of days clients use the exercise facility per week is 2.86 days. This value provides a central point around which the data clusters. It tells us that, on average, a client uses the facility almost 3 days a week. However, to understand the spread of the data, we need to calculate the standard deviation, which will tell us how much individual data points deviate from this mean.

The standard deviation is a critical measure of dispersion in statistics. It quantifies the amount of variation or spread in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range. To calculate the standard deviation for our exercise facility usage data, we'll use the following formula for a sample standard deviation:

s = √[ Σ(f * (X - μ)²) / (n - 1) ]

Where:

  • s is the sample standard deviation
  • X is the number of days per week
  • f is the frequency for each X
  • μ is the mean (2.86 days)
  • n is the sample size (100)
  • Σ means "sum of"

Let's break down the calculation step by step:

  1. Calculate the difference between each X and the mean (X - μ):
    • 0 - 2.86 = -2.86
    • 1 - 2.86 = -1.86
    • 2 - 2.86 = -0.86
    • 3 - 2.86 = 0.14
    • 4 - 2.86 = 1.14
    • 5 - 2.86 = 2.14
    • 6 - 2.86 = 3.14
  2. Square these differences ( (X - μ)² ):
    • (-2.86)² = 8.1796
    • (-1.86)² = 3.4596
    • (-0.86)² = 0.7396
    • (0.14)² = 0.0196
    • (1.14)² = 1.2996
    • (2.14)² = 4.5796
    • (3.14)² = 9.8596
  3. Multiply each squared difference by its frequency (f * (X - μ)²):
    • 2 * 8.1796 = 16.3592
    • 13 * 3.4596 = 44.9748
    • 30 * 0.7396 = 22.188
    • 29 * 0.0196 = 0.5684
    • 11 * 1.2996 = 14.2956
    • 8 * 4.5796 = 36.6368
    • 7 * 9.8596 = 69.0172
  4. Sum these products (Σ(f * (X - μ)²)): 16.3592 + 44.9748 + 22.188 + 0.5684 + 14.2956 + 36.6368 + 69.0172 = 204.04
  5. Divide by (n - 1): 204.04 / (100 - 1) = 204.04 / 99 = 2.061
  6. Take the square root: √2.061 ≈ 1.436

Therefore, the standard deviation (s) is approximately 1.436 days. This value tells us how much the individual data points typically deviate from the mean of 2.86 days. Now that we have both the mean and standard deviation, we can answer the question about the number that is 1.5 standard deviations from the mean.

Now that we have calculated the mean (μ = 2.86 days) and the standard deviation (s ≈ 1.436 days), we can determine the number that is 1.5 standard deviations from the mean. This involves calculating two values: one that is 1.5 standard deviations above the mean and one that is 1.5 standard deviations below the mean. These values will give us a range within which a certain proportion of the data falls, according to the empirical rule (or 68-95-99.7 rule) in a normal distribution. However, since we are only asked for "the number", we will calculate the value above the mean.

To find the number 1.5 standard deviations above the mean, we use the following formula:

Value = μ + (1.5 * s)

Plugging in our values:

Value = 2.86 + (1.5 * 1.436) Value = 2.86 + 2.154 Value = 5.014

So, the number that is 1.5 standard deviations above the mean is approximately 5.014 days. This means that a client who uses the facility about 5 days a week is using it significantly more than the average client in our sample. Understanding these deviations from the mean helps the exercise facility identify clients with different usage patterns, which can inform targeted interventions or programs. For instance, clients using the facility much less than the average might benefit from encouragement or tailored exercise plans, while those using it much more could be candidates for advanced programs or special recognition.

In conclusion, our statistical analysis of exercise facility usage provides valuable insights into client behavior. We found that the mean usage is approximately 2.86 days per week, indicating that, on average, clients use the facility close to 3 days a week. The standard deviation of approximately 1.436 days tells us about the variability in usage patterns. Furthermore, we determined that the number 1.5 standard deviations above the mean is approximately 5.014 days, highlighting clients who use the facility significantly more often than the average.

These findings have several implications for the exercise facility:

  • Targeted Programs: Understanding the mean and standard deviation allows the facility to create targeted programs. For example, clients who use the facility less frequently might benefit from introductory programs or personalized encouragement, while those who use it more frequently could be offered advanced classes or opportunities to become involved in facility events.
  • Resource Allocation: Knowing the typical usage patterns can help the facility allocate resources effectively. For instance, if most clients use the facility during specific times of the day or week, staffing and equipment can be adjusted to meet the demand.
  • Client Engagement: Identifying clients who deviate significantly from the average usage can inform engagement strategies. Clients using the facility much less than average might need additional support or motivation, while those using it much more could be valuable advocates for the facility.
  • Benchmarking: Comparing these usage statistics with those of other facilities can provide valuable benchmarks for performance and identify areas for improvement.

By leveraging this statistical analysis, the exercise facility can make data-driven decisions to improve its services, enhance client engagement, and optimize resource allocation. This approach not only benefits the facility but also contributes to the overall health and well-being of its clients. Moving forward, the facility can continue to collect and analyze data to track trends, measure the impact of interventions, and adapt its strategies to meet the evolving needs of its client base.