Question

Twenty patients with severe epilepsy were observed for 8 weeks. The following are the numbers of major seizures suffered by each patient during the observation period: ${}^{56}$ $$\begin{array}{llllllllll}5& 0& 9& 6& 0& 0& 5& 0& 6& 1\\ 5& 0& 0& 0& 0& 7& 0& 0& 4& 7\end{array}$$ (a) Is the distribution of seizures bimodal? Justify your answer. (b) Calculate the SD of seizures. (c) What percentage of the seizures is within $1\mathrm{SD}$ of the mean? (d) What percentage of the seizures is within 2 SDs of the mean?

Short answer

The distribution may be bimodal if there are two peaks in the frequency distribution. The standard deviation of seizures will be calculated using the derived mean and variance. The percentages of seizures within 1 SD and 2 SDs of the mean will be calculated based on the seizure count within those ranges.

Step-by-step solution

Construct a Frequency Distribution

Count the number of occurrences of each number of seizures and create a frequency distribution chart. The chart should list the numbers of seizures and their corresponding frequencies.

Analyze the Distribution for Bimodality

Examine the frequency distribution to determine if there are two distinct 'peaks' or high-frequency values, which would indicate a bimodal distribution.

Calculate the Mean Number of Seizures

Add up all the seizure numbers and divide by the total number of patients (20) to find the mean (average) number of seizures per patient.

Calculate the Standard Deviation (SD)

First, find the variance by subtracting the mean from each individual's number of seizures, squaring those differences, summing them up, and then dividing by the total number of patients minus one. Take the square root of variance to determine the standard deviation.

Find the Number of Seizures within 1 SD of the Mean

Count the number of seizures that fall between mean-SD and mean+SD, then calculate this as a percentage of the total number of seizures to answer part (c).

Find the Number of Seizures within 2 SDs of the Mean

Count the number of seizures that fall between mean-2SD and mean+2SD, then calculate this as a percentage of the total number of seizures to answer part (d).

Key concepts

Bimodal Distribution

Understanding the concept of a bimodal distribution is crucial when analyzing data with potential dual peaks or modes. Consider a situation where seizure counts are being measured across a group of individuals. Bimodal distribution occurs when the data shows two different peaks or high-frequency areas. This might suggest the existence of two different sub-groups or causes within the dataset, such as patients with different seizure triggers or severities.

For example, if many patients experience either very few or many seizures, a bimodal distribution might become evident, with peaks at both ends of the frequency chart. To identify if a distribution is bimodal, you need to create a frequency distribution chart and look for two distinct areas where seizure counts have higher occurrences. This analysis is part of the step-by-step solution process, aiding in visualizing whether the distribution of seizures amongst the observed patients exhibits this bimodal nature.

Standard Deviation

Another fundamental concept in statistics is standard deviation (SD), which measures the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean (average) of the set, whereas a high standard deviation indicates that the values are spread out over a wider range.

In the context of epilepsy patients, the standard deviation of seizure counts can help us understand the consistency of seizures among the patients. To calculate SD, as outlined in the solution steps, it involves finding the mean, computing the variance by summing the squared differences from the mean, and then taking the square root of the variance. The resulting SD value is crucial for determining the variability in seizure frequency and answering more detailed questions about the distribution, such as the percentage of seizures occurring within one or two standard deviations from the mean.

Frequency Distribution Chart

A frequency distribution chart serves as a visual representation of how often each value in a set of data occurs. When constructing this chart, you list the distinct values of seizures (for example, the number of seizures each patient had during an observation period) and their corresponding frequencies.

A well-constructed frequency distribution chart allows for immediate insight into the data’s shape and spread. For instance, you can easily spot whether the data leans towards certain values or if a bimodal distribution is present. In the epilepsy study, the chart could reveal concentration points of seizure counts, helping to understand the distribution and aid in further statistical calculations, such as finding the mean and standard deviation. When applied correctly, this chart is an invaluable tool for uncovering patterns in complex datasets and facilitating data-driven decisions in medical studies.