Question

Dopamine is a chemical that plays a role in the transmission of signals in the brain. A pharmacologist measured the amount of dopamine in the brain of each of seven rats. The dopamine levels (nmoles/g) were as follows: \({ }^{41}\) $$ \begin{array}{lllllll} 6.8 & 5.3 & 6.0 & 5.9 & 6.8 & 7.4 & 6.2 \end{array} $$ (a) Calculate the mean and SD. (b) Determine the median and the interquartile range. (c) Replace the observation 7.4 by 10.4 and repeat parts (a) and (b). Which of the descriptive measures display robustness and which do not?

Short answer

Mean=6.34, SD=0.73; Median=6.2, IQR=0.95. With 10.4: Mean=6.7, SD=1.72; Median=6.2, IQR=1.05. The median and IQR are robust; the mean and SD are not.

Step-by-step solution

- Calculate the Mean

To calculate the mean, sum up all the values and divide by the number of values. Mean = (6.8 + 5.3 + 6.0 + 5.9 + 6.8 + 7.4 + 6.2) / 7.

- Calculate the Standard Deviation (SD)

First, find the variance by calculating the squared difference from the mean for each value, summing those squared differences, and then dividing by the number of values. Then, take the square root of the variance to get the SD.

- Determine the Median

Order the values from least to greatest and pick the middle number. If there's an even number of observations, average the two middle numbers.

- Calculate the Interquartile Range (IQR)

After arranging the values in increasing order, the IQR is found by subtracting the first quartile (25th percentile) from the third quartile (75th percentile).

- Repeat Calculation with Adjusted Values

Replace 7.4 with 10.4 and calculate the mean and SD. Evidently, calculate the new median and IQR.

- Assess Robustness

Compare the results from parts (a) and (b) before and after the change to determine which measures are robust to outliers and which are not.

Key concepts

Calculating Mean and Standard Deviation

Understanding how to calculate the mean and standard deviation of a data set is fundamental in statistical data analysis. The mean, often referred to as the average, is a measure of the central tendency of a data set. To calculate the mean for a given set of numbers, sum all the values, then divide by the number of values you have.

For instance, with dopamine levels in rats measured in nmoles/g, if we sum up 6.8, 5.3, 6.0, 5.9, 6.8, 7.4, and 6.2, we get a total of 44.4. We then divide by the number of rats, 7, to get a mean value.The standard deviation (SD), on the other hand, captures the variability or spread of the data around the mean. First, we calculate the variance by finding the squared difference from the mean for each value, summing those, and then dividing by the number of values. The SD is the square root of the variance, giving us a measure of how spread out the numbers in our data set are.

• A small SD indicates that the data points tend to be close to the mean.
• A large SD indicates that the data points are spread out over a larger range of values.

Calculating the SD is crucial, especially in experimental science, to understand the reliability and precision of the experiment.

Determining Median and Interquartile Range

The median is the middle value of a data set that's been ordered from least to greatest. It's a robust measure of central tendency, especially useful when dealing with skewed distributions or outliers.

To find the median, one must order the data points and pick the middle one. In the case of an even number of observations, the median is the average of the two middle numbers. For the dopamine data set with seven values, the median would be the fourth value when arranged in ascending order.The interquartile range (IQR), on the other hand, measures the spread of the middle 50% of the data. It is determined by subtracting the 25th percentile (first quartile) from the 75th percentile (third quartile). The IQR is particularly important as it is less affected by outliers than the range, making it a more reliable measure of spread for skewed distributions or data sets with outliers. Understanding the IQR helps in identifying how spread out the middle portion of the data is and is also used in creating boxplots for visual data analysis.

Robustness of Statistical Measures

The concept of robustness in statistical measures is about the insensitivity of these measures to outliers or to small departures from assumed distributional forms. In essence, a robust statistical measure is one that is not significantly affected by outliers or non-normality in the data set.

Robust measures like the median and IQR are preferred when the data includes outliers or non-normal distributions since they can provide a better sense of the central tendency and variability within the main bulk of the data. For example, the median is solely determined by the middle value(s) and not by the extreme values, unlike the mean which includes all data points in its calculation and can therefore be skewed by outliers. Similarly, the IQR focuses only on the middle 50% of the data, ignoring the tails where outliers reside. This resilience makes such measures especially valuable in the field of pharmacology where outliers can be caused by abnormalities in an individual subject's response to a substance.

Outliers in Data Sets

Outliers are data points that differ significantly from other observations in a data set. They can skew the results of an analysis and can be caused by variability in the measurement or possibly by experimental error. Identifying outliers is crucial for any robust statistical analysis, as they can affect the mean and standard deviation significantly.

When outliers are present, the mean becomes a less reliable measure of central tendency, as it gets pulled towards the outlier. In our example with dopamine levels, when a value of 7.4 is replaced with 10.4, the mean increases, showing its vulnerability to extreme values. The standard deviation, which measures the spread of data, will also increase because one data point is now much farther from the new mean.

• Identifying outliers is typically one of the first steps in data analysis.
• It's crucial to determine if they are due to genuine variation or if they are a result of data recording errors or other issues.

Appropriate handling of outliers is important to ensure the accuracy of statistical conclusions. Depending on the context, outliers may be removed, adjusted, or left in the analysis with caution noted.