The weights (in pounds) of 30 newborn babies are listed below. Find Q1: 5.5, 5.7, 5.8, 6.0, 6.1, 6.1, 6.3, 6.4, 6.5, 6.6, 6.7, 6.7, 6.7, 6.9, 7.0, 7.0, 7.0, 7.1, 7.2, 7.2, 7.4, 7.5, 7.7, 7.7, 7.8, 8.0, 8.1, 8.1, 8.3, 8.7.
Mathematics · College · Thu Feb 04 2021
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To find the first quartile (Q1) of the given data set, which consists of the weights of 30 newborn babies, we need to determine the value below which 25% of the data falls. Since the data appears to be in ascending order, we can proceed with finding Q1 directly.
The first quartile Q1 is the value at the 25th percentile of the data set. When dealing with a dataset with an even number of observations, Q1 is found between the \( \frac{N}{4} \)th and \( (\frac{N}{4} + 1) \)th data points, where N is the number of data points. For 30 observations, this would be between the 7.5th and the 8th data point when you line them up in order.
Here are the steps to find Q1:
1. Multiply the total number of data points (30) by 0.25 (25%) to find the rank of the first quartile. 2. \( 30 \times 0.25 = 7.5 \). Since we cannot have half an observation, we take the average of the 7th and 8th values in the ordered dataset for our Q1.
The 7th value in the data set is 6.3, and the 8th value is 6.4. Now, calculate the average of these two values to find Q1:
\[ Q1 = \frac{6.3 + 6.4}{2} = \frac{12.7}{2} = 6.35 \]
Therefore, the first quartile (Q1) of the weights of the newborn babies is 6.35 pounds.
Extra: When we talk about quartiles, we are referring to values that divide a set of observations into four equal parts. The dataset is first sorted in ascending order. There are three quartiles: the first quartile (Q1), the second quartile (Q2), and the third quartile (Q3).
- Q1 is the median of the first half of the dataset, which corresponds to the 25th percentile. - Q2 is the median of the entire dataset and corresponds to the 50th percentile. - Q3 is the median of the second half of the dataset, which corresponds to the 75th percentile.
Quartiles are useful measures of spread that help to understand the distribution of data and to detect outliers. They are commonly used in boxplots, where Q1 and Q3 form the edges of the box, and Q2 is marked by a line within the box. Understanding and interpreting quartiles is a key concept in statistics, and it’s often taught in middle school and high school math classes.