Graphical Analysis In Exercises 9–12, determine whether the approximate shape of the distribution in the histogram is symmetric, uniform, skewed left, skewed right, or none of these. Justify your answer.

Verified step by step guidance

1. Observe the histogram provided. Note the frequency of data points across the intervals on the x-axis (52.5, 62.5, 72.5, 82.5). The bars increase in height as we move to the right, peaking at the interval 82.5, and then slightly decreasing.
2. Determine the symmetry of the distribution. A symmetric distribution would have a mirror-like shape around its center. In this case, the histogram is not symmetric because the left side is much shorter than the right side.
3. Evaluate whether the distribution is uniform. A uniform distribution would have bars of approximately equal height across all intervals. This histogram does not exhibit uniformity as the heights of the bars vary significantly.
4. Assess whether the distribution is skewed left or skewed right. A skewed left distribution has a longer tail on the left side, while a skewed right distribution has a longer tail on the right side. Here, the histogram has a longer tail on the left side, indicating it is skewed left.
5. Conclude the shape of the distribution. Based on the observations, the histogram is skewed left. This conclusion is justified by the longer tail on the left side and the peak occurring towards the right side of the graph.

Key Concepts

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into intervals (bins) and the frequency of data points within each interval is represented by the height of bars. It helps visualize the shape of the data distribution, making it easier to identify patterns such as central tendency, variability, and skewness.

Distribution Shape

The shape of a distribution refers to the way data points are spread across the range of values. Common shapes include symmetric (equal on both sides), uniform (equal frequency across intervals), skewed left (tail on the left side), and skewed right (tail on the right side). Understanding the shape is crucial for interpreting data characteristics and making statistical inferences.

Skewness

Skewness measures the asymmetry of a distribution around its mean. A distribution is skewed left if it has a longer tail on the left side, indicating that most data points are concentrated on the right. Conversely, a right-skewed distribution has a longer tail on the right. Identifying skewness helps in understanding the nature of the data and can influence the choice of statistical methods for analysis.