Total Blood Cholesterol Level Measured In 21 Adults

The Total Blood Cholesterol Level Was Measured For 21adults

The total blood cholesterol level was measured for each of 21 adults. Here are the 21 measurements (in mg/dL): 129, 144, 159, 162, 163, 168, 183, 184, 184, 185, 186, 195, 196, 197, 198, 207, 209, 221, 222, 234, 246.

(a) Which measures of central tendency do not exist for this data set? Choose all that apply. Mean Median Mode None of these measures

In this data set, all three primary measures of central tendency—mean, median, and mode—exist. The mean can be calculated as the sum of all measurements divided by 21, which is clearly defined, and numerical. The median, being the middle value when data are ordered, exists; the data are ordered from smallest to largest, and the 11th value (186) is the median. The mode, which is the most frequently occurring value, exists; in this data set, 184 appears twice, making it the mode.

Therefore, none of these measures do not exist. The correct choice is: None of these measures.

(b) Suppose that the measurement 129 (the smallest measurement in the data set) were replaced by 52. Which measures of central tendency would be affected by the change? Choose all that apply.

Replacing 129 with 52 would impact the measures of central tendency. Specifically, the mean would decrease significantly, because the sum of all measurements would reduce, lowering the average. The median could potentially change if the new value influences the middle position—since 52 is smaller than the original lowest value, it would likely move the median slightly down or remain the same depending on the adjusted ordering. However, the mode would not be affected because 52 appears only once, replacing a value that was also unique. Ultimately, the measures affected include the mean and possibly the median, but not the mode.

Correct choices: Mean, Median.

(c) Suppose that, starting with the original data set, the smallest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply.

Removing the smallest value (129) from the original data set would affect the mean, increasing it slightly if the new smallest value is higher than 129, but since 129 is the smallest, removing it causes the data to shift, and the mean may change accordingly. The median could remain the same or shift if the number of data points changes and the position of the middle value shifts. As for the mode, since 129 was only present once, it would not have been the mode, so removing it would not affect the mode.

In summary, removing the smallest value will likely affect the mean and possibly the median, but not the mode.

Correct choices: Mean, Median.

(d) Which of the following best describes the distribution of the original data? Choose only one.

Looking at the data, the smallest measurement is 129 and the largest is 246. The bulk of the data seems centered around the 180s and 190s, with some higher values extending into the 220s and 240s, and an outlier at 52 before replacement. The distribution appears fairly symmetric, as the measurements rise evenly on both sides, with no sudden skew. The outlying low value (before replacement, 129) doesn't significantly skew the distribution, but if replaced with 52, the distribution might appear skewed, leaning toward the positive side.

Based on the original data, which contains a fairly symmetric spread without significant skewness, the best description is: Roughly symmetrical.

Paper For Above instruction

The analysis of blood cholesterol levels provides valuable insights into health trends among adults and is fundamental in epidemiology and public health research. To interpret such data accurately, understanding measures of central tendency and distribution patterns is essential. In this paper, we examine the measures of central tendency—mean, median, and mode—and how they are affected by data modifications such as values being replaced or removed. Additionally, we classify the distribution shape of the dataset and explore its implications for data interpretation.

Understanding Measures of Central Tendency

The three primary measures of central tendency—mean, median, and mode—serve as vital statistics that summarize the characteristics of a dataset. The mean provides the arithmetic average, which is sensitive to extremely high or low values, known as outliers. The median indicates the middle value when data are ordered and is robust against outliers, thus offering a reliable central point in skewed distributions. The mode reflects the most frequently occurring value, highlighting commonality within the dataset.

In the case of the presented blood cholesterol data, all three measures exist. The mean can be calculated using the sum of the measurements divided by the number of observations, which is 21. The median is determined by arranging data in order and choosing the middle value, which in this case, is the 11th observation at 186 mg/dL. The mode appears as 184, which occurs twice, marking the most common measurement. Since none of the measures are undefined or nonexistent, the correct answer to the first question is that none are missing.

Impact of Data Modification on Measures of Central Tendency

Changing a data point can influence the measurements of central tendency differently. Replacing the smallest measurement, 129 mg/dL, with a lower value like 52 mg/dL, significantly impacts the mean because the total sum decreases, reducing the average. Similarly, the median, which depends on the middle data points, may also shift downward because the lowest value is now even lower, potentially changing the ordered sequence and the middle position. The mode remains unaffected unless the new value, 52, becomes the most frequent, which is not the case here, as it appears only once.

Removing the smallest value (129 mg/dL) affects the dataset differently. Since this value is unique, its removal does not alter the mode. However, the mean could change slightly depending on the new sum of measurements, and the median could shift if the removal affects the position of the middle value due to change in the data count. Typically, the median is stable under such removal unless the middle position's value is altered, which is likely here. Overall, both the mean and median can be affected by such a change, but the mode is unaffected.

Distribution Shape and Its Implications

The distribution of the blood cholesterol data appears roughly symmetric, with scores dispersed around the central values in a balanced manner. The data range from 129 to 246, with the majority clustering in the 180s and 190s. There is no evident skewness in the original dataset because it doesn't favor one tail over the other significantly. The outlier at 129, which was substantially lower than the rest, was an outlier that might have skewed the distribution if included as an outlier but is not a part of the original distribution pattern. After considering the overall spread, the distribution can be described as roughly symmetrical, indicating a balanced variation of cholesterol levels among the sample population.

Conclusion

Understanding measures of central tendency and distribution shapes enables researchers and health professionals to interpret cholesterol data effectively. Recognizing how data modifications like value replacements and removals influence these measures helps in accurate analysis and decision-making. Describing the distribution provides additional context for understanding the underlying health profile of the population studied. Overall, this analysis underscores the importance of statistical literacy in epidemiological research and health assessment.

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