Question

Exercise 22 looked at distances PGA golfers can hit the ball. The standard deviation of these average drive distances is \(9.3\) yards, and the quartiles are \(Q_{1}=282\) yards and \(Q_{3}=294\) yards. a) Write a sentence or two describing the spread in distances based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

Short answer

Quartiles reveal the middle 50% of variation, while standard deviation measures average distance from the mean. Both have limitations if data is skewed.

Step-by-step solution

Understanding Quartiles

Quartiles divide a data set into four equal parts. The first quartile \(Q_1\) is the median of the lower half of the data and the third quartile \(Q_3\) is the median of the upper half of the data set. Here, \(Q_1 = 282\) yards and \(Q_3 = 294\) yards. This means that 50% of the data lies between 282 yards and 294 yards, indicating that half of the golfers have driving distances within this 12-yard range.

Understanding Standard Deviation

Standard deviation is a measure of the dispersion or spread of a data set. It shows how much variation there is from the average (mean). In this case, the standard deviation is \(9.3\) yards, indicating that the drive distances typically vary by this amount from the mean drive distance.

Describing Spread Using Quartiles

Using the quartiles, the interquartile range \( (IQR) \) can be calculated as \(Q_3 - Q_1 = 294 - 282 = 12\) yards. This tells us that the middle 50% of drive distances are spread over a range of 12 yards.

Describing Spread Using Standard Deviation

The standard deviation of \(9.3\) yards tells us about the average dispersion around the mean. This implies that individual drive distances typically deviate from the mean by around 9.3 yards.

Evaluating Quartiles Description

Quartiles provide a measure of spread but might not effectively summarize the spread if the data is heavily skewed. They give no information about data points outside the middle 50%.

Evaluating Standard Deviation Description

Standard deviation assumes a normal distribution of the data. If the data is not symmetrically distributed, the description given by the standard deviation might not effectively capture its variability. Variability may be higher or lower if the data is skewed, which could distort the interpretation based on standard deviation.

Key concepts

Quartiles

Quartiles are a helpful way to understand how a data set is distributed by dividing it into four equal parts. This helps us see where most of the data points lie. The first quartile (\(Q_1\)) and the third quartile (\(Q_3\)) are key markers in this division.

In our golfer example, \(Q_1\) is 282 yards and \(Q_3\) is 294 yards. This means that 50% of the golfers can hit between 282 and 294 yards. The range between \(Q_1\) and \(Q_3\), known as the interquartile range (IQR), is 12 yards. This IQR tells us how concentrated or spread out the middle half of the data is. So, the golfers' drives are moderately spread within this middle range.

Using quartiles, we can see exactly where the bulk of data points lie, but quartiles might not give us information on how extreme values are distributed outside the middle 50%. When the data is skewed, either positively or negatively, depending solely on quartiles could be misleading. It's essential to consider other measures to get the full picture of data spread.

Standard Deviation

Standard deviation is a statistical tool that helps us understand the average distance between each data point and the mean of the data set. It gives us valuable information about the data's spread concerning its average value. A small standard deviation means the data points are close to the mean, whereas a larger one indicates a wider spread around the mean.

In our golf ball distance example, the standard deviation is 9.3 yards. This figure tells us that the typical distance that a golfer can hit deviates from the average by about 9.3 yards. It's an effective way to summarize the data's variability, making it easier to grasp how consistent or variable the distances are around the average drive distance.

Understanding and using standard deviation can be quite potent, but it does rely on an assumption. It assumes the data follows a normal distribution, meaning data clusters evenly around the mean. When the data isn't symmetric, and is instead heavily skewed, the standard deviation's impression of variability might not accurately reflect the true spread of the data set. In such situations, relying solely on the standard deviation could lead to an incomplete understanding.

Data Spread

Data spread is the range through which our data points are distributed across our dataset. It's important to grasp this concept to fully comprehend the variability and reliability of data.

• The quartiles show us where most of the data lies to divide the data into four parts. The IQR tells us how wide the center part of the data is.
• The standard deviation, on the other hand, provides a measure of how data points spread around the mean, giving insight into the consistency or variability from the average.

Data spread is crucial because it can affect conclusions drawn from data analysis. A data set with less spread might mean more consistent data, whereas larger spread could imply more variability, potentially reducing predictability.

Relying on multiple ways to assess spread, such as using both quartiles and standard deviation together, helps cross-checks results and provides a more comprehensive picture of data behavior. It's always key to consider not just one but multiple measures to understand fully how data behaves, and this includes plotting the data visually for any skewness or outliers. Such practices ensure that interpretations are not merely dependent on one form of analysis, especially in cases where data distribution is unknown or thought to be skewed. This holistic approach ultimately leads to better, more robust statistical conclusions.