91Ó°ÊÓ

Age at diagnosis for each of 20 patients under treatment for meningitis was given in the paper "Penidlin in the Treatment of Meningitis" (journal of the American Medical Association [1984]: \(1870-1874\) ). The ages (in years) were as follows: $$ \begin{array}{lllllllllll} 8 & 25 & 19 & 23 & 20 & 69 & 18 & 21 & 18 & 20 & 18 \\ 0 & 18 & 19 & 28 & 17 & 18 & 18 & & & & \end{array} $$ \(\begin{array}{ll}18 & 18 \\ 18 & 20\end{array}\) a. Calculate the values of the sample mean and the standard deviation. b. Calculate the \(10 \%\) trimmed mean. How does the value of the trimmed mean compare to that of the sample mean? Which would you recommend as a measure of center? Explain. c. Compute the upper quartile, the lower quartile, and the interquartile range. d. Are there any mild or extreme outliers present in this data set? e. Construct the boxplot for this data set.

Step by step solution

01

Calculate The Sample Mean

To get the sample mean, sum all the ages given and then divide by the total number of samples (20).

02

Calculate The Standard Deviation

To calculate the standard deviation, follow these steps: 1. Subtract the mean from each observation (this gives the 'deviation' of each observation). 2. Square each deviation (this gives the 'squared deviation'). 3. Calculate the mean of these squared deviations. 4. Find the square root of the mean squared deviation.

03

Calculate the 10% Trimmed Mean

Trimmed mean requires removing the top and bottom 10% of the data. Here, remove 2 observations from each end (total 20 observations), sum the remaining data, and then divide by the number of remaining observations.

04

Calculate Upper Quartile, Lower Quartile and Interquartile Range

Firstly, arrange the data in ascending order. The lower quartile (first quartile or Q1) is the median of the lower half of the data (excluding the median if even examples), the upper quartile (third quartile or Q3) is the median of the upper half of the data. The interquartile range (IQR) is the range within which the central 50% of the data values fall, and is calculated as Q3 - Q1.

05

Identify Outliers

Outliers are data points that are much different from other observations. An outlier can be either mild or extreme. A data point is considered a:1. Mild outlier if it is between 1.5 and 3 times the IQR above Q3 or below Q1.2. Extreme outlier if it is more than 3 times the IQR above Q3 or below Q1.

06

Construct a Boxplot

A boxplot involves drawing a box from Q1 to Q3, with a line indicating the median. Whiskers are drawn from the box to the smallest and largest data points that are within 1.5 IQR from Q1 and Q3 respectively. Points outside the range of the whiskers are considered outliers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean

In descriptive statistics, the sample mean represents the central location of a data set. It is essentially the average of all the data points you have collected. To find the sample mean, you add up all the values and divide the sum by the number of values in the data set. For instance, if you have ages, like the ones provided for patients diagnosed with meningitis, you sum all those ages and then divide by 20, which is the total number of patients you are considering here.
This statistic gives a basic idea of the average age, but it's sensitive to extreme values or outliers in the dataset, which means a few very young or very old ages could skew the mean. So while it's a useful measure, it's important to know it's not always representative of typical values in your data.

Standard Deviation

The standard deviation is a measure of how spread out the numbers are in your data set. It's a critical part of descriptive statistics because it gives you a sense of the variability or consistency among the numbers you are working with.
To calculate standard deviation, you begin by figuring out deviations from the mean for each data point — this means subtracting the mean from each individual data point. Next, square those deviations to remove any negative signs and accentuate larger differences. Then you compute the average of these squared deviations. Finally, take the square root of this average to return to the original unit of measurement, giving you the standard deviation.
A larger standard deviation indicates more spread in the data, whereas a smaller standard deviation indicates that the data points are closer to the mean.

Trimmed Mean

A trimmed mean is designed to give you a more robust measure of central tendency, especially when your data set includes outliers that could skew your results. In a 10% trimmed mean, you remove the smallest 10% and the largest 10% of your data.
For example, in a data set of 20 ages, you would eliminate the 2 smallest and 2 largest ages before calculating the mean of the remaining 16 values. This trimmed mean provides a central value that is less affected by extremes.
It can often be more representative of the data set when there are outliers, as it focuses on the core of your data distribution rather than extremes.

Quartiles

Quartiles split your data into four equal parts and help in understanding its distribution. These include the lower quartile (Q1), the median (Q2), and the upper quartile (Q3).

• Q1 (lower quartile) is the midpoint of the first half of your data once sorted in ascending order,
• Q2 (median) is the middle value of the data set,
• Q3 (upper quartile) is the midpoint of the second half of the data set.

The interquartile range (IQR) is calculated by subtracting Q1 from Q3 and represents the range of the middle 50% of your data. This can be useful for identifying variability among the middle proportion of your data, which is untouched by the extremes. The IQR is also crucial for identifying outliers.

Boxplot

A boxplot, also known as a box-and-whisker plot, is a graphical representation of the distribution of data based on a five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
These plots are constructed by drawing a 'box' from Q1 to Q3, indicating the IQR, and a line within the box showing the median. Lines ('whiskers') extend from the box to the smallest and largest values within 1.5 times the IQR from Q1 and Q3, respectively. Outliers, defined as being outside this range, are marked separately.
Boxplots are a simple, effective way to visualize the range, central value, and variability of data. They also make it easy to spot outliers and compare distributions across different data sets at a glance.