Question

SGOT is an enzyme that shows elevated activity when the heart muscle is damaged. In a study of 31 patients who underwent heart surgery, serum levels of SGOT were measured 18 hours after surgery. 30 The mean was $49.3\mathrm{U}/\mathrm{l}$ and the standard deviation was $68.3\mathrm{U}/\mathrm{l}$. If we regard the 31 observations as a sample from a population, what feature of the data would cause one to doubt that the population distribution is normal?

Short answer

The standard deviation being much larger than the mean suggests high dispersion and potential skewness, casting doubt on the assumption of normality.

Step-by-step solution

Examine the Sample Mean and Standard Deviation

Look at the given sample mean ($49.3\mathrm{U}/\mathrm{l}$) and standard deviation ($68.3\mathrm{U}/\mathrm{l}$) to assess the central tendency and variability in the data.

Consider the Implications of the Standard Deviation Relative to the Mean

Assess whether the standard deviation is large relative to the mean. A large standard deviation in comparison to the mean suggests a high level of dispersion, which could indicate a skewed distribution.

Evaluate the Likelihood of a Normal Distribution

Since the standard deviation is much larger than the mean, one could suspect the data may not be symmetrically distributed. This could be a sign of a skewed distribution which may be inconsistent with the assumptions of normality.

Consider Additional Statistical Tests

To confirm doubts about normality, one could perform a statistical test for normality such as the Shapiro-Wilk test or examine a histogram or Q-Q plot for the sample data.

Key concepts

Sample Mean

In statistics, the sample mean is a critical measure that represents the average value of a data sample. It is calculated by summing all the individual observations and dividing by the number of observations in the sample. For instance, in a study of SGOT enzyme levels in patients after heart surgery, the sample mean was reported as 49.3 U/l. This mean serves as an indicator of the central point around which the individual SGOT levels are dispersed. It's essential to note that while the sample mean provides a sense of the central tendency, it doesn't reflect potential asymmetry or outliers in the data distribution.

To improve understanding, remember that the sample mean is like finding the balance point of a see-saw if each data value had the same weight. It tells us what the typical value is in our sample.

Standard Deviation

The standard deviation is a statistic that quantifies the amount of variability or spread in a set of data. The larger the standard deviation, the more scattered the data points are from the mean. In the case of the SGOT levels with a standard deviation of 68.3 U/l, we can infer that there's considerable variability in the patient enzyme levels post-surgery.

Think of standard deviation as the average distance of the data points from the mean—like how far each person lives from the center of a town. A small standard deviation would mean everyone lives close together, while a large value indicates they live much further apart.

Skewed Distribution

A skewed distribution is a scenario where data points are not symmetrically distributed around the mean. There are two types of skew: positive (right-skewed) where most data are to the left of the mean, or negative (left-skewed) where the reverse is true. Skewness can be a sign that the assumption of normality might not hold. For example, if the SGOT levels are significantly higher for a few patients, this might produce a positively skewed distribution.

To picture skewness, imagine a competition where everyone throws balls at a target. Most people hit near the center, but a few throw much further. If the throws are gathered into a pile, they won't form a neat stack but instead will have a long tail where those further throws landed.

Statistical Tests for Normality

Statistical tests for normality help determine whether a data set is likely to have come from a normally distributed population. Examples include the Shapiro-Wilk test and the Kolmogorov-Smirnov test. When the p-value from the test is less than a predefined significance level (usually 0.05), it suggests that the data do not follow a normal distribution. In practice, running these tests would provide a more definitive check on the normality of our SGOT level data beyond just assessing the sample mean and standard deviation.

A test for normality is akin to a litmus test for the symmetry of the data. It gives a more rigorous mathematical basis to either support or question our preliminary observations of the sample's distribution.

Central Tendency

Central tendency is a statistical measure to determine the center of a data distribution. The most common measures are the mean, median, and mode. The mean is sensitive to extreme values or outliers, which can pull it in the direction of the skew. The median, being the middle value when ordered, and the mode, being the most frequent value, are less affected by outliers and can be more representative of 'typical' values in a skewed distribution. For the SGOT data, the mean alone may not give a complete picture of central tendency if the distribution is skewed.

To understand central tendency better, consider it as the most representative score or value for a group. If we were summarizing the performance of a class on a test, we could use the average score as the central tendency, even though it might not capture performance variations or outliers.

Variability

Variability refers to the degree to which scores in a data set differ from each other and from their mean. Variability is essential for understanding the spread and distribution of the data. Standard deviation and variance are the most commonly used measures to quantify variability. The high standard deviation in the SGOT data suggests that the patients' enzyme levels vary widely from the mean, indicating that a simple average may not reliably summarize the data set.

When thinking about variability, imagine a classroom where students' heights are measured. If every student is of similar height, variability would be low. But if students' heights vary greatly, variability would be high, indicating a diverse mix of statures.

Histogram Analysis

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into intervals (bins), and the frequency of data points within each bin is depicted by the height of the bar. Histogram analysis allows us to visually assess the shape, central tendency, and variability of the data. If the histogram of SGOT levels showed a bell-shaped curve, it would suggest a normal distribution. Conversely, a histogram that is not symmetrical and has a pronounced tail might indicate skewness, challenging the normality assumption.

The analogy is to think of a histogram like a city skyline. Just as different buildings have different heights and widths, a histogram's bars give us a visual summary of how many data points fall within each range and how they are distributed across the entire sample.

Q-Q Plot

A Q-Q (quantile-quantile) plot is a visual tool to compare the distribution of a dataset against a theoretical distribution, such as the normal distribution. Points on a Q-Q plot lie on a straight line if the sample distribution matches the theoretical distribution. Deviations from this line indicate departures from normality. A Q-Q plot for the SGOT level data would show us how closely the patients' post-surgery enzyme levels follow a normal distribution.

To grasp the concept, imagine plotting the height of children against adults on a graph. If children grow consistently into the adult heights, the points will line up straight. Any deviations would show growth patterns not following the expected trend, much like a Q-Q plot reveals about data normality.