Question

Physicians measured the concentration of calcium (nM) in blood samples from 38 healthy persons. The data are listed as follows: \(^{65}\) $$ \begin{array}{rrrrrrr} 95 & 110 & 135 & 120 & 88 & 125 \\ 112 & 100 & 130 & 107 & 86 & 130 \\ 122 & 122 & 127 & 107 & 107 & 107 \\ 88 & 126 & 125 & 112 & 78 & 115 \\ 78 & 102 & 103 & 93 & 88 & 110 \\ 104 & 122 & 112 & 80 & 121 & 126 \\ 90 & 96 & & & & \end{array} $$ Calculate appropriate measures of the center and spread of the distribution. Describe the shape of the distribution and any unusual features in the data.

Short answer

To analyze the distribution, list the data in ascending order, calculate the mean, median, and mode for central tendency, compute the range, variance, and standard deviation for spread, and evaluate skewness and outliers to outline the shape and unusual features.

Step-by-step solution

Organize the Data

List the data in ascending order to make it easier to calculate measures of central tendency and spread.

Calculate Measures of Central Tendency

Compute the mean, median, and mode of the data set to determine the center of the distribution.

Calculate Measures of Spread

Calculate the range, variance, and standard deviation to understand the spread of the data.

Describe the Shape and Features of the Distribution

Analyze the data's skewness and look for any outliers to describe the shape and features of the distribution.

Key concepts

Measures of Central Tendency

Understanding the central point around which data values are concentrated is essential in biostatistics. The measures of central tendency—mean, median, and mode—help identify this focal point. The mean is the arithmetic average of all data points, calculated by summing up all the values and dividing by the number of values. In the context of calcium concentration in blood samples, physicians would sum up all measurements and divide by 38 to find the mean concentration.

The median represents the middle value when data is organized in ascending or descending order. If there is an odd number of observations, it is simply the middle one; if even, it's the average of the two central numbers. This is less affected by extreme values or outliers than the mean. The mode is the most frequently occurring value in a data set. Although less commonly used for numerical data, it can be informative, especially if the data set has a value that repeats significantly more than others.

Measures of Spread

Once we know where the center of the data lies, the next step is to understand the variability or spread of the data. This includes how far apart data points are from each other and from the center. The range is the simplest measure of spread, calculated by subtracting the smallest value from the largest. However, the range doesn't provide information about the distribution of values between the extremes.

To get a deeper insight, we calculate the variance and standard deviation. The variance is the average of squared differences from the mean, providing a sense of overall data variability. The formula is \( \sigma^2 = \frac{\sum(x_i - \bar{x})^2}{N} \), where \( x_i \) is each value, \( \bar{x} \) is the mean, and \( N \) is the number of observations. The standard deviation is the square root of variance and provides a measure of spread in the same units as the data, which makes it more interpretable.

Data Distribution Analysis

Data distribution analysis involves more than just pinpointing the center and spread; it includes visualizing and interpreting the shape and patterns within the data. Skewness indicates the symmetry of the distribution, while kurtosis reflects the 'peakedness'. A perfect normal distribution, for example, would have a skewness close to zero and a kurtosis value indicating a bell-shaped curve.

Data can also be represented through histograms or box plots, with the latter highlighting the median, quartiles, and potential outliers. Analyzing the blood calcium levels involves looking for any asymmetry or unusual features such as gaps or clusters, which can signify interesting biological phenomena or potential errors in data collection.

Calcium Concentration in Blood

Calcium plays a vital role in the human body, and its concentration in the blood is tightly regulated. Normal values are critical for bone health, muscle function, and cellular communication. For physicians, understanding the typical range and distribution of blood calcium levels in healthy individuals is crucial for diagnosing and managing diseases like hypercalcemia or hypocalcemia.

By applying biostatistical methods to analyze calcium concentrations, researchers can establish reference ranges, discern patterns related to age or gender, and identify potential health risks. Additionally, any anomalies in the distribution might suggest areas for further medical investigation.