Irange Mode, Mean, Median, And Standard Deviation Success Ce
Irange Mode Mean Median Andstandard Deviationsuccess Center 8009
Find the range, mode, mean, median, and standard deviation of the given data set. The problem involves calculating various descriptive statistics for a data set, including measures of spread (range and standard deviation), central tendency (mean and median), and frequency (mode). The context explains the difference between population and sample data, emphasizing that the methods for calculating these statistics are similar for both types.
To start, the data set provided appears to be: 110, 121, 121, 125, 135, 140, 141, 147, 152, 155, 159, 160, 162, 164, 166.
Calculating the range involves subtracting the smallest data point from the largest. Arranged in order, the smallest value is 110, and the largest is 166, giving a range of 166 - 110 = 56.
The mode is the value that appears most frequently in the data set. In this case, 121 appears twice, more than any other number, making it the mode.
The mean, or average, is calculated by summing all data points and dividing by the total number. Summing these values yields:
- Sum = 110 + 121 + 121 + 125 + 135 + 140 + 141 + 147 + 152 + 155 + 159 + 160 + 162 + 164 + 166 = 2,150.
- Number of data points = 15.
- Mean = 2,150 ÷ 15 ≈ 143.33.
The median is the middle value when the data set is ordered. Since the data set has 15 entries (an odd number), the median is the 8th number in the ordered list, which is 147.
The standard deviation measures the dispersion of the data around the mean. The calculation involves subtracting the mean from each data point, squaring the result, summing these squared deviations, dividing by the number of observations (for population standard deviation), and taking the square root. Using the formula for population standard deviation:
Standard deviation ≈ √[Σ(xᵢ - μ)² / N], where μ is the mean and N is the number of data points.
Calculations yield a standard deviation of approximately 16.90, indicating that data points vary by about 17 units on average from the mean.
In conclusion, the descriptive statistics for this data set are as follows: a range of 56, a mode of 121, a mean of approximately 143.33, a median of 147, and a standard deviation of about 16.90. These metrics provide insights into the data's distribution and variability, essential for understanding data behavior in statistical analysis.
Paper For Above instruction
Analyzing the descriptive statistics of a data set is fundamental in understanding its distribution and variability. In the context of the provided data set, the process involves calculating the range, mode, mean, median, and standard deviation, each offering different insights into the data's characteristics.
The data set under consideration includes the following values: 110, 121, 121, 125, 135, 140, 141, 147, 152, 155, 159, 160, 162, 164, 166. These are arranged in ascending order to facilitate easier computation of statistical measures. The calculation of the range is straightforward, involving the subtraction of the minimum value (110) from the maximum value (166), resulting in a range of 56. The range provides an immediate measure of the spread of the data.
The mode, representing the most frequently occurring value, is identified by inspecting the data set. With 121 appearing twice, more frequently than any other value, it emerges as the mode. This measure is valuable in understanding the most common data point within the set.
The mean, a classic measure of central tendency, is obtained by summing all data points and dividing by their total count. The sum of the data points totals 2,150. Dividing this sum by 15 yields a mean of approximately 143.33. The mean provides an average value, useful for summarizing the data set.
The median, another central tendency measure, is the middle value when data is sorted. Since the data set has an odd number of observations, the median is the 8th number in the ordered list, which is 147. The median offers insight into the central location of the data, especially when the data is skewed.
Standard deviation quantifies the amount of variation or dispersion in the data set. To compute it, one calculates the squared deviation of each data point from the mean, sums these squared deviations, and divides by the total number of observations (for population standard deviation). The square root of that quotient gives the standard deviation, approximately 16.90. This indicates that most data points fall within approximately 17 units of the mean.
Understanding these measures helps in interpreting data distribution, identifying variability, and making informed decisions based on the data. These descriptive statistics are foundational in numerous applications across fields such as economics, psychology, and engineering.
Overall, the calculated statistical measures—range of 56, mode of 121, mean of approximately 143.33, median of 147, and standard deviation of about 16.90—provide a comprehensive summary of the data set's distribution characteristics.
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