Calculate The Mean, Median, Mode, Range, And Variance For The Data Set: 67, 70, 76, 77, 79, 84, 87, 89, 89, 89, 90, 95, 95, 100, 121.
In data analysis, understanding the characteristics of a dataset is crucial for drawing meaningful conclusions. Measures of central tendency and dispersion are fundamental tools that provide valuable insights into the distribution and variability of data. This article will delve into these measures, exploring their significance and how they can be applied to interpret data effectively. We will specifically analyze the dataset: 67, 70, 76, 77, 79, 84, 87, 89, 89, 89, 90, 95, 95, 100, 121, to illustrate the calculation and interpretation of these measures.
Unveiling Central Tendency: Mean, Median, and Mode
Central tendency measures aim to identify the center or typical value within a dataset. The three primary measures of central tendency are the mean, median, and mode. Each measure offers a unique perspective on the data's central location, and understanding their differences is essential for comprehensive data analysis.
The Mean: The Arithmetic Average
The mean, often referred to as the average, is calculated by summing all the values in a dataset and dividing by the total number of values. It is a widely used measure that provides a balanced representation of the data's central point. In the given dataset (67, 70, 76, 77, 79, 84, 87, 89, 89, 89, 90, 95, 95, 100, 121), the mean is calculated as follows:
Mean = (67 + 70 + 76 + 77 + 79 + 84 + 87 + 89 + 89 + 89 + 90 + 95 + 95 + 100 + 121) / 15 = 87.2
The mean of 87.2 kg suggests that the average value in the dataset is around 87.2 kg. However, the mean can be sensitive to extreme values or outliers, which can distort its representation of the central tendency. In this dataset, the value 121 is noticeably higher than the other values, which may influence the mean.
The Median: The Middle Ground
The median is the middle value in a dataset when it is arranged in ascending order. It is a robust measure of central tendency that is not affected by outliers. To find the median, we first need to sort the dataset: 67, 70, 76, 77, 79, 84, 87, 89, 89, 89, 90, 95, 95, 100, 121. Since there are 15 values in the dataset, the median is the 8th value, which is 89 kg.
The median of 89 kg indicates that half of the values in the dataset are below 89 kg, and half are above it. Compared to the mean, the median provides a more stable measure of central tendency when outliers are present.
The Mode: The Most Frequent Value
The mode is the value that appears most frequently in a dataset. It is a useful measure for identifying the most common observation. In the given dataset, the value 89 appears three times, which is more than any other value. Therefore, the mode is 89 kg.
The mode of 89 kg suggests that this value is the most typical or representative value in the dataset. Datasets can have multiple modes (bimodal or multimodal) or no mode at all if all values occur with the same frequency.
Understanding Dispersion: Range, Variance, and Standard Deviation
Dispersion measures the spread or variability of data points in a dataset. It provides insights into how much the data deviates from the central tendency. The common measures of dispersion include range, variance, and standard deviation.
Range: The Spread of Data
The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. In our dataset, the maximum value is 121, and the minimum value is 67. Therefore, the range is:
Range = 121 - 67 = 54 kg
The range of 54 kg gives a quick overview of the data's spread, but it is highly sensitive to outliers. A single extreme value can significantly inflate the range, making it less representative of the overall variability.
Variance: Measuring the Average Squared Deviation
The variance is a more comprehensive measure of dispersion that quantifies the average squared deviation of each data point from the mean. It provides a sense of how much the data points are scattered around the mean. To calculate the variance, we first find the difference between each value and the mean, square these differences, sum them up, and then divide by the number of values minus 1 (for sample variance) or the number of values (for population variance). The formula for sample variance (s²) is:
s² = Σ(xi - x̄)² / (n - 1)
Where:
- xi represents each individual value in the dataset
- x̄ represents the sample mean
- n represents the number of values in the sample
Let's calculate the variance for our dataset:
- Calculate the deviations from the mean (87.2):
- 67 - 87.2 = -20.2
- 70 - 87.2 = -17.2
- 76 - 87.2 = -11.2
- 77 - 87.2 = -10.2
- 79 - 87.2 = -8.2
- 84 - 87.2 = -3.2
- 87 - 87.2 = -0.2
- 89 - 87.2 = 1.8
- 89 - 87.2 = 1.8
- 89 - 87.2 = 1.8
- 90 - 87.2 = 2.8
- 95 - 87.2 = 7.8
- 95 - 87.2 = 7.8
- 100 - 87.2 = 12.8
- 121 - 87.2 = 33.8
- Square the deviations:
- (-20.2)² = 408.04
- (-17.2)² = 295.84
- (-11.2)² = 125.44
- (-10.2)² = 104.04
- (-8.2)² = 67.24
- (-3.2)² = 10.24
- (-0.2)² = 0.04
- (1.8)² = 3.24
- (1.8)² = 3.24
- (1.8)² = 3.24
- (2.8)² = 7.84
- (7.8)² = 60.84
- (7.8)² = 60.84
- (12.8)² = 163.84
- (33.8)² = 1142.44
- Sum the squared deviations:
- Σ(xi - x̄)² = 408.04 + 295.84 + 125.44 + 104.04 + 67.24 + 10.24 + 0.04 + 3.24 + 3.24 + 3.24 + 7.84 + 60.84 + 60.84 + 163.84 + 1142.44 = 2496.32
- Divide by (n - 1) = 15 - 1 = 14:
- s² = 2496.32 / 14 ≈ 178.31
The variance is approximately 178.31 kg². However, the variance is in squared units, making it less intuitive to interpret directly. This is where the standard deviation comes in.
Standard Deviation: The Square Root of Variance
The standard deviation is the square root of the variance. It measures the average distance of data points from the mean in the original units of measurement. It is a widely used measure of dispersion due to its ease of interpretation. The formula for sample standard deviation (s) is:
s = √s²
Using the variance calculated above (178.31 kg²), the standard deviation is:
s = √178.31 ≈ 13.35 kg
The standard deviation of approximately 13.35 kg indicates that, on average, data points in the dataset deviate from the mean by about 13.35 kg. A higher standard deviation suggests greater variability, while a lower standard deviation indicates that data points are clustered closer to the mean.
Confidence Intervals: Estimating Population Parameters
A confidence interval provides a range of values within which a population parameter, such as the population mean, is likely to lie with a certain level of confidence. It is a crucial tool in inferential statistics, allowing us to make generalizations about a population based on a sample.
To calculate a confidence interval, we need the sample mean, sample standard deviation, sample size, and the desired level of confidence. The formula for a confidence interval for the population mean (μ) when the population standard deviation is unknown is:
Confidence Interval = x̄ ± (tα/2 * (s / √n))
Where:
- x̄ is the sample mean
- tα/2 is the critical t-value for the desired confidence level and degrees of freedom (n - 1)
- s is the sample standard deviation
- n is the sample size
To construct a confidence interval for our dataset, we need to choose a confidence level. Let's assume a 95% confidence level. With a sample size of 15, the degrees of freedom are 15 - 1 = 14. Using a t-table or a statistical calculator, the critical t-value (t0.025, 14) for a 95% confidence level is approximately 2.145.
Now, we can calculate the confidence interval:
Confidence Interval = 87.2 ± (2.145 * (13.35 / √15)) Confidence Interval = 87.2 ± (2.145 * (13.35 / 3.873)) Confidence Interval = 87.2 ± (2.145 * 3.447) Confidence Interval = 87.2 ± 7.39
The 95% confidence interval is (87.2 - 7.39, 87.2 + 7.39), which is approximately (79.81, 94.59).
This means that we are 95% confident that the true population mean lies within the range of 79.81 kg to 94.59 kg.
Conclusion: Applying Measures for Data Interpretation
Measures of central tendency, dispersion, and confidence intervals are essential tools for data analysis and interpretation. The mean, median, and mode provide insights into the central location of data, while range, variance, and standard deviation quantify the spread or variability. Confidence intervals allow us to estimate population parameters with a specified level of confidence. By understanding and applying these measures, we can gain a deeper understanding of datasets and make informed decisions based on data-driven insights. In the context of the dataset analyzed (67, 70, 76, 77, 79, 84, 87, 89, 89, 89, 90, 95, 95, 100, 121), we observed that the mean is 87.2 kg, the median is 89 kg, and the mode is 89 kg, indicating a central tendency around 87-89 kg. The range of 54 kg, standard deviation of approximately 13.35 kg, and the 95% confidence interval of (79.81, 94.59) kg provide further insights into the data's dispersion and the likely range for the population mean. These measures collectively offer a comprehensive understanding of the dataset's characteristics, enabling meaningful interpretation and decision-making.
Central Tendency, Dispersion, Confidence Interval, Mean, Median, Mode, Range, Variance, Standard Deviation, Data Analysis, Statistics, Population Parameter, Sample, Dataset.