Question

Use these \(n=15\) measurements to answer the questions in Exercises 11 and 12. 53,61,58,56,58,60,54,54,62,58,60,58,56,56,58 Are the data skewed right, skewed left, or symmetric? Draw a dotplot to confirm your answer.

Short answer

Data set: 54, 54, 53, 56, 56, 56, 58, 58, 58, 58, 60, 60, 61, 62 Answer: The data set is symmetric.

Step-by-step solution

Arrange the data in ascending order

Arrange the given data set in ascending order. 54, 54, 53, 56, 56, 56, 58, 58, 58, 58, 60, 60, 61, 62

Calculate Mean, Median, and Mode

Calculate the mean (average), median (middle value), and mode (most frequent value) of the data: Mean = \(\frac{53+61+\cdots+56+58}{15}=\frac{864}{15}=57.6\) Median = Middle value when data is ordered = 58 Mode = Most frequent value = 58

Determine the skewness based on Mean, Median, and Mode

Determine if the data is skewed right, skewed left, or symmetric based on the following conditions: - If Mean > Median > Mode, then the data is skewed right. - If Mean < Median < Mode, then the data is skewed left. - If Mean = Median = Mode, then the data is symmetric. In our case, Mean < Median = Mode (57.6 < 58 = 58), the data is symmetric.

Draw a Dotplot

To confirm our answer, let's draw a dotplot of the data. A dotplot is a simple graphical representation that displays each data point as a dot along a number line. 53: * 54: * * 56: * * * 58: * * * * * 60: * * 61: * 62: * By observing the dotplot, we can see that it is relatively symmetric, supporting our previous conclusion that the data is symmetric.

Key concepts

Mean

The mean is commonly referred to as the average. It is calculated by adding up all the numbers in a data set and then dividing the sum by the number of values in the set. Here's how you can visualize the calculation:

• Sum of all measurements: 53 + 61 + 58 + 56 + 58 + 60 + 54 + 54 + 62 + 58 + 60 + 58 + 56 + 56 + 58 = 864
• Number of measurements: 15
• Mean = \(\frac{864}{15} = 57.6\)

The mean provides a central value of the data, giving a quick idea of the overall level of the measurements. However, it can be affected by extremely high or low values, known as outliers.

Median

The median is the middle value of a data set when arranged in ascending order. If there is an odd number of values, it is the single middle number. If there is an even number, the median is the average of the two central numbers.

• Arrange: 53, 54, 54, 56, 56, 56, 58, 58, 58, 58, 60, 60, 61, 62
• Count the numbers, find the middle: 58 (8th number in the ordered list)

In this example, with 15 data points, the 8th number is the median. The median is less sensitive to outliers, making it a useful measure of central tendency when the data includes skewed values.

Mode

The mode represents the most frequently occurring value within a data set. It is useful to understand which number appears most often. In the context of our exercise:

• Identify frequencies: 53 (1), 54 (2), 56 (3), 58 (5), 60 (2), 61 (1), 62 (1)
• Mode = 58 (most frequent, appears 5 times)

The mode helps identify the most typical value, which can be particularly useful in understanding distribution, especially in non-numerical data.

Skewness

Skewness describes the symmetry of a data distribution. A symmetrical distribution has the mean, median, and mode all the same, and any deviation indicates skewness.

• If Mean > Median > Mode, it's skewed right.
• If Mean < Median < Mode, it's skewed left.
• If Mean = Median = Mode, the distribution is symmetric.

In this example, 57.6 (Mean) < 58 (Median and Mode), the data appears symmetric. Observations can also be confirmed visually through plots like dotplots, which can illustrate skewness efficiently by showing data symmetry or imbalance.