When drawing box and whisker plots, how do you know when you no longer need to draw the whiskers? For instance, in the following data points: 3,42,45,55,56,58,and 84, you do not need whiskers. However, in the data points: 12,35,37,41,53,58,59,91, you do need whiskers. Thanks!

When drawing box and whisker plots, the decision about whether to include whiskers depends on the presence of any outlier values in the data set. Outliers are values that are significantly smaller or larger than the majority of the data points.

To determine if whiskers are needed, you need to calculate the lower quartile (Q1), the upper quartile (Q3), and the interquartile range (IQR) of the dataset. The quartiles divide the data into four equal parts, with Q1 being the value below which 25% of the data falls and Q3 being the value below which 75% of the data falls. The IQR is the range between Q1 and Q3.

To find outliers, you can use the following formula:
1. Calculate Q1 - 1.5 * IQR
2. Calculate Q3 + 1.5 * IQR

Any data point that is smaller than the Q1 - 1.5 * IQR or larger than Q3 + 1.5 * IQR is considered an outlier.

If there are no outliers in the dataset, you do not need to draw whiskers. In your example, the first set of data points (3, 42, 45, 55, 56, 58, and 84) does not have any outliers, so you wouldn't draw whiskers.

However, in the second set of data points (12, 35, 37, 41, 53, 58, 59, 91), you need to calculate Q1, Q3, and IQR. Suppose Q1 = 35, Q3 = 58, and IQR = Q3 - Q1 = 58 - 35 = 23. Then, using the outlier formula:

Q1 - 1.5 * IQR = 35 - 1.5 * 23 = 35 - 34.5 = 0.5
Q3 + 1.5 * IQR = 58 + 1.5 * 23 = 58 + 34.5 = 92.5

Since one data point, 91, is larger than Q3 + 1.5 * IQR, it is considered an outlier. Therefore, you would draw whiskers in this case.

Remember, the whiskers on a box and whisker plot extend to the lowest non-outlier value and the highest non-outlier value. Any data points beyond that are considered outliers and are represented as individual points outside the whiskers.