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Chapter 2: Problem 34

Density curves Sketch a density curve that might describe a distribution that has a single peak and is skewed to the left.

Short Answer

Expert verified
Draw a density curve with a peak on the right and a longer left tail.

Step by step solution

01

Understanding the Problem

To solve this problem, we need to visualize a density curve that represents a unimodal (single peak) distribution where the longer tail stretches towards the left, indicating that it's skewed to the left. Skewness describes the asymmetry of the distribution around its peak.
02

Identifying Key Characteristics

A density curve for a skewed left distribution typically has a peak located towards the right side of the graph and a long tail extending towards the left side. This graph indicates that the majority of the data values are higher, with fewer lower values spreading out to the left.
03

Sketching the Density Curve

Begin by drawing a horizontal axis (x-axis) and a vertical axis (y-axis). Mark a single peak near the right side of the x-axis. From the peak, the curve should descend steeply on the right side and gradually taper off to the left, creating a left tail.
04

Finalizing the Sketch

Ensure the area under the curve equals 1, as this is a typical requirement for density curves. Check that the peak is placed more rightward compared to the left tail, clearly demonstrating the left skewness of the distribution. The curve should never dip below the x-axis, as probabilities cannot be negative.

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Key Concepts

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

Skewed Distribution
A skewed distribution occurs when a dataset's values are not symmetric around the peak or the mean, as a result, the data 'leans' to one side.
It is important to understand that skewness doesn't merely mean an extended tail. It indicates a shift in data concentration. Depending on this data concentration and the direction of the tail, we can determine whether it's left-skewed or right-skewed:
  • **Right Skewed (positive skew):** The tail is on the right side. This implies most of the data points are concentrated on the low-value end, with the higher values extending out towards the tail.
  • **Left Skewed (negative skew):** The tail is on the left side. Here, most data points gather at the higher end while the fewer lower values stretch out towards the tail.
Recognizing a skewed distribution is often done by visually analyzing a density curve, noticing where the peak and tails lie relative to the center line.
Unimodal Distribution
A distribution is labeled as unimodal when it has one, distinct peak or mode, occurring most frequently in the dataset. This is a vital concept in understanding the overall shape of data distributions in statistics.
To better comprehend unimodal distributions, keep in mind the following:
  • A unimodal distribution means there is only one "hump" or peak in the curve.
  • The most common example of a unimodal distribution is the normal distribution, which is perfectly symmetric. However, unimodal distributions can also be skewed.
Unimodality is an essential characteristic when analyzing skewness, as the single peak indicates the prominent point of central tendency.
Left Skewness
Left skewness, often referred to as negatively skewed, occurs when the tail stretches towards the left.
In a left-skewed distribution, the mean is usually less than the median because the lower values in the tail pull the mean down.
Key attributes of left skewness include:
  • The density curve's peak is positioned closer to the upper end of the value range.
  • The longer tail indicates that there are relatively few very low values.
  • This distribution is common in real-world datasets, such as income levels where most people earn lower to mid-range salaries and fewer earn at the extreme low end.
Left skewness is crucial when interpreting statistical data as it affects how averages are understood.

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Most popular questions from this chapter

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