Calculating Standard Deviation And Variance For The Dataset 110, 125, 245, 165, 201, 147
To determine the standard deviation for the given dataset, which includes the numbers 110, 125, 245, 165, 201, and 147, we will follow a step-by-step process. Understanding standard deviation is crucial in statistics as it measures the dispersion or spread of a dataset around its mean. A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation suggests that the data points are clustered closer to the mean. This measure is extensively used across various fields, from finance to science, to understand data variability and make informed decisions.
First, we need to calculate the mean of the dataset. The mean is the average of all the numbers. We add all the data points together and divide by the number of data points. In this case, we add 110, 125, 245, 165, 201, and 147, and then divide by 6, since there are six numbers in the dataset. The formula for the mean (μ) is:
μ = (Σx) / n
Where Σx is the sum of all the data points and n is the number of data points. Once we have the mean, we proceed to the next step, which involves finding the variance. Variance is another key statistical measure that quantifies the spread of data points in a dataset. It is calculated as the average of the squared differences from the mean. Each data point's deviation from the mean is squared to ensure that all differences are positive, thus preventing positive and negative deviations from canceling each other out. This gives a clearer picture of the overall variability in the data.
Next, we calculate the variance. The variance is the average of the squared differences from the mean. For each number in the dataset, we subtract the mean and then square the result. We then add up all these squared differences and divide by the number of data points minus 1 (this is known as the sample variance, which provides a better estimate of the population variance when dealing with a sample). The formula for the sample variance (s^2) is:
s^2 = Σ(x - μ)^2 / (n - 1)
Where x represents each data point, μ is the mean, and n is the number of data points. This calculation provides a comprehensive measure of how much the individual data points deviate from the average. The higher the variance, the more spread out the data is from the mean. This step is vital in understanding the distribution and consistency of the data, setting the stage for the final calculation of standard deviation.
Finally, we calculate the standard deviation. The standard deviation is the square root of the variance. It provides a measure of the spread of the data in the same units as the original data, making it more interpretable than the variance. The formula for the standard deviation (s) is:
s = √s^2
Taking the square root of the variance transforms the value back into the original units of measurement, providing an intuitive understanding of the data's dispersion. This final calculation is essential for comparing different datasets and drawing meaningful conclusions about the data's characteristics. By following these steps, we can accurately determine the standard deviation for the dataset, which is a crucial metric in statistical analysis.
Let’s apply these steps to our dataset. First, calculate the mean:
Mean (μ) = (110 + 125 + 245 + 165 + 201 + 147) / 6 = 993 / 6 = 165.5
Now that we have the mean, we can calculate the variance. We find the squared difference of each data point from the mean, sum them up, and divide by n-1 (which is 5 in this case):
Variance (s^2) = [(110 - 165.5)^2 + (125 - 165.5)^2 + (245 - 165.5)^2 + (165 - 165.5)^2 + (201 - 165.5)^2 + (147 - 165.5)^2] / 5
Variance (s^2) = [(-55.5)^2 + (-40.5)^2 + (79.5)^2 + (-0.5)^2 + (35.5)^2 + (-18.5)^2] / 5
Variance (s^2) = [3080.25 + 1640.25 + 6320.25 + 0.25 + 1260.25 + 342.25] / 5
Variance (s^2) = 12643.5 / 5 = 2528.7
Finally, we calculate the standard deviation by taking the square root of the variance:
Standard Deviation (s) = √2528.7 ≈ 50.29
Therefore, the standard deviation for the dataset is approximately 50.29. This value gives us a quantitative measure of the spread of the data points around the mean. A standard deviation of 50.29 indicates a moderate degree of variability in the dataset, helping us understand how the individual data points are distributed relative to the average. In conclusion, the standard deviation for the dataset 110, 125, 245, 165, 201, 147 is approximately 50.29, making option 50.29 the correct answer.
In this section, we will focus on calculating the variance for the same dataset: 110, 125, 245, 165, 201, and 147. The variance is a crucial statistical measure that quantifies the spread of data points in a dataset. It is defined as the average of the squared differences from the mean. This metric is essential in various fields, including finance, engineering, and science, as it provides a clear understanding of the data's variability. A higher variance indicates that the data points are more spread out, while a lower variance suggests they are clustered more closely around the mean.
To begin, we must first calculate the mean of the dataset. As established in the previous section, the mean is the average of all data points. We add up all the numbers in the dataset and divide by the total number of data points. For our dataset, this involves adding 110, 125, 245, 165, 201, and 147, and then dividing by 6 (since there are six numbers). The formula for the mean (μ) is expressed as:
μ = (Σx) / n
Where Σx represents the sum of all data points, and n is the number of data points. This initial step is crucial because the mean serves as the central reference point from which all deviations are measured. Once the mean is calculated, we can proceed to the next step, which involves determining the squared differences from the mean. This process lays the foundation for understanding how individual data points vary in relation to the average value.
Next, we calculate the squared differences from the mean. For each data point, we subtract the mean and then square the result. Squaring the differences ensures that all values are positive, preventing negative deviations from canceling out positive ones. This step is vital because it gives a clear indication of the magnitude of each data point's deviation from the mean, irrespective of direction. The formula to calculate the squared difference for each data point (x) is:
(x - μ)^2
We perform this calculation for each number in the dataset. This process provides a set of values that represent the squared deviations from the mean, which are then used to calculate the variance. Squaring the differences highlights larger deviations more significantly, making the variance a sensitive measure of data dispersion. This step is a critical component in understanding the overall spread and consistency of the data.
Finally, we calculate the variance. This is done by summing up all the squared differences and dividing by the number of data points minus 1 (n - 1). The use of (n - 1) instead of n is known as Bessel's correction, which provides an unbiased estimate of the population variance when using a sample. This is particularly important when working with smaller datasets, as it helps to correct for the underestimation of the population variance that can occur when dividing by n. The formula for the sample variance (s^2) is:
s^2 = Σ(x - μ)^2 / (n - 1)
Where x represents each data point, μ is the mean, and n is the number of data points. The variance provides a single value that summarizes the overall dispersion of the dataset. A higher variance suggests that the data points are more spread out from the mean, while a lower variance indicates that the data points are clustered more closely around the mean. This final calculation is essential for understanding the distribution and variability of the data, making it a key metric in statistical analysis. Understanding the variance helps in making informed decisions and drawing meaningful conclusions from the data.
Let's apply these steps to our dataset. We've already calculated the mean in the previous section:
Mean (μ) = 165.5
Now, we calculate the squared differences from the mean:
(110 - 165.5)^2 = (-55.5)^2 = 3080.25 (125 - 165.5)^2 = (-40.5)^2 = 1640.25 (245 - 165.5)^2 = (79.5)^2 = 6320.25 (165 - 165.5)^2 = (-0.5)^2 = 0.25 (201 - 165.5)^2 = (35.5)^2 = 1260.25 (147 - 165.5)^2 = (-18.5)^2 = 342.25
Next, we sum up the squared differences:
Σ(x - μ)^2 = 3080.25 + 1640.25 + 6320.25 + 0.25 + 1260.25 + 342.25 = 12643.5
Finally, we calculate the variance by dividing the sum of the squared differences by n - 1 (which is 5 in this case):
Variance (s^2) = 12643.5 / 5 = 2528.7
Therefore, the variance for the dataset 110, 125, 245, 165, 201, 147 is 2528.7. This value represents the average of the squared differences from the mean, providing a quantitative measure of the dispersion of the data points. A variance of 2528.7 indicates a significant degree of variability in the dataset. Thus, the variance for the given dataset is 2528.7, making option 2528.08 the closest and correct answer.
In summary, the variance calculation for the dataset involves finding the mean, calculating the squared differences from the mean, summing these squared differences, and dividing by (n - 1). This process yields a value that quantifies the spread of the data, which is essential for statistical analysis and informed decision-making.