Question

Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or skewed? Explain why. a) The number of speeding tickets each student in the senior class of a college has ever had. b) Players' scores (number of strokes) at the U.S. Open golf tournament in a given year. c) Weights of female babies born in a particular hospital over the course of a year. d) The length of the average hair on the heads of students in a large class.

Short answer

(a) Unimodal, skewed right; (b) Unimodal, symmetric; (c) Unimodal, symmetric; (d) Bimodal, possibly skewed.

Step-by-step solution

Assessing Variable (a)

Consider the distribution of the number of speeding tickets among college seniors. Most students are likely to have none or only a few speeding tickets, making the distribution unimodal with a peak at low counts. As some students may have many tickets, there could be a long tail to the right, suggesting the distribution is skewed to the right.

Evaluating Variable (b)

Examine the distribution of players' scores at the U.S. Open. Player performance usually centers around a few common score values, creating a unimodal distribution. Since golf scoring often results in a range of scores around a mean, with deviations distributed fairly evenly, the distribution is likely symmetric or slightly skewed based on outlier performances.

Analyzing Variable (c)

Investigate the weights of female babies born in a hospital. Baby weights typically follow a normal distribution, centering around an average with fewer instances of exceptionally high or low weights, suggesting a unimodal and symmetric distribution.

Understanding Variable (d)

Look at the lengths of hair among students in a large class. This is likely to be bimodal because many students may have either very short or longer hair, representing two distinct peaks corresponding to different common hair lengths. The distribution could be skewed depending on cultural or gender factors influencing hair length.

Key concepts

Skewness

Skewness describes the asymmetry of a data set's distribution. When we look at a distribution, we often want to know whether it is symmetrical or if it leans more to one side. Knowing about skewness helps us understand where most values lie and whether there are outliers affecting our distribution.

• A right-skewed distribution has a tail that extends to the right. This indicates that there are some exceptionally high values pulling the mean to the right of the median. An example from the exercise is the number of speeding tickets amongst students where most may have few tickets, but a few with many tickets cause right skewness.
• In contrast, a left-skewed distribution turns its tail to the left due to low outlier values pulling the mean to the left of the median.

Understanding skewness is crucial because it affects statistical analyses and insights drawn from the data. For example, right-skewed data like that from the speeding ticket scenario can affect calculations of central tendency and variability.

Unimodal Distribution

A unimodal distribution is one with a single peak or mode. This is often the most common type of distribution, thanks to real-world instances where data naturally centers around a particular value.

• Player scores in a golf tournament can typically present a unimodal distribution. Scores center around common outcomes due to similar performance levels among participants.
• Similarly, baby weights are expected to display a unimodal pattern. Most newborns' weights cluster around a healthy average, with fewer being drastically over or underweight.

The shape of a unimodal distribution might be symmetrical, like a bell curve or, occasionally skewed due to outliers, but it remains characterized by a single peak.
Recognizing a distribution as unimodal helps predict where most data points fall, which aids in understanding the norm within the dataset.

Bimodal Distribution

A bimodal distribution, unlike unimodal, has two distinct peaks or modes. This pattern suggests the data group around two different centers or values, which can arise from mixtures of different populations within the dataset.

• The case of hair lengths among students in a large class often leads to a bimodal distribution. This occurs because there are typically two primary groups: one preferring shorter hair and another longer hair. Each group creates its peak within the distribution.
• Bimodal distributions are insightful as they highlight the presence of distinct sub-groups within the data, hinting at diverse patterns rather than a single norm.

Recognizing bimodal distributions is essential for effectively analyzing data, as it can impact how conclusions are drawn. When planning surveys or studies, understanding whether a bimodal nature exists can direct more tailored questions towards different population subsets.