Final answer:
A histogram visually represents data distribution by showing the frequency of occurrences within different ranges of values. When analyzing Shakespeare's texts, the histogram and smooth curve illustrate the distribution of word lengths, indicating patterns such as normal distribution, skewness, and modality.
Step-by-step explanation:
To explain how a histogram represents the distribution of lengths of words in Shakespeare's plays, one must understand that a histogram is a graphical representation of data. When constructing a histogram, we divide the range of data into intervals, known in this context as 'bins' or 'classes.' Each bin represents a range of word lengths, and the height of the bar corresponds to the frequency of words falling into that range. For instance, a tall bar at the interval representing 4-letter words would indicate that there are many words of this length in Shakespeare’s text.
The shape of the distribution will show whether the lengths of words are evenly spread out (uniform), clustered around a central value (normal distribution), or skewed toward shorter or longer words. This shape is highlighted by drawing a smooth curve through the midpoints of the tops of the bars. The curve can provide insight on the prevalence of different word lengths used by Shakespeare, indicating a certain linguistic style or textual complexity.
After sketching the histogram, one could describe, for example, that the distribution has a peak (or mode) at a particular word length, indicating a preference for words of that length. Or, if the curve is skewed to the right, it would show a propensity for using shorter words more frequently than longer words in his plays. Therefore, a histogram and the accompanying smooth curve not only quantify the lengths of words used but also illustrate the overall linguistic pattern within the text.