Question

A class of fourth graders takes a diagnostic reading test, and the scores are reported by reading grade level. The 5-number summaries for the 14 boys and 11 girls are shown: $$ \begin{array}{llllll} \text { Boys: } & 2.0 & 3.9 & 4.3 & 4.9 & 6.0 \\ \text { Girls: } & 2.8 & 3.8 & 4.5 & 5.2 & 5.9 \end{array} $$ a) Which group had the highest score? b) Which group had the greater range? c) Which group had the greater interquartile range? d) Which group's scores appear to be more skewed? Explain. e) Which group generally did better on the test? Explain. f) If the mean reading level for boys was \(4.2\) and for girls was \(4.6\), what is the overall mean for the class?

Short answer

a) Boys; b) Boys; c) Girls; d) Boys, skewed right; e) Girls; f) 4.38.

Step-by-step solution

Identify the Highest Scores from Each Group

The highest score for boys is given by the maximum value from their 5-number summary, which is 6.0. Meanwhile, the highest score for girls is 5.9. Compare these two values to determine which group had the highest score.

Calculate the Range for Each Group

The range is determined by subtracting the minimum value from the maximum value. For boys, calculate the range as \(6.0 - 2.0 = 4.0\). For girls, the range is \(5.9 - 2.8 = 3.1\). Compare these results to find the group with the greater range.

Calculate the Interquartile Range (IQR) for Each Group

The interquartile range is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). For boys, the IQR is \(4.9 - 3.9 = 1.0\). For girls, the IQR is \(5.2 - 3.8 = 1.4\). Compare these IQRs to determine which group has the greater interquartile range.

Analyze Skewness of Each Group's Scores

Skewness is indicated by the position of the median between Q1 and Q3 and the comparison of Q3 - median and median - Q1. For boys, the median is 4.3, closer to Q3 (4.9). Therefore, \(4.9 - 4.3 = 0.6\) and \(4.3 - 3.9 = 0.4\), suggesting mild right skewness. For girls, the median is 4.5, \(5.2 - 4.5 = 0.7\) and \(4.5 - 3.8 = 0.7\), suggesting a symmetric distribution. The boys' scores appear more skewed right.

Compare Overall Performance

To determine which group generally did better, compare the medians and means. Boys have a median of 4.3 and a mean of 4.2, whereas girls have a median of 4.5 and a mean of 4.6, suggesting that girls generally performed better.

Calculate the Overall Mean

The overall mean can be calculated using the formula \(\text{Overall mean} = \frac{(n_1 \times \text{Mean}_1) + (n_2 \times \text{Mean}_2)}{n_1 + n_2}\), where \(n_1 = 14\) and \(\text{Mean}_1 = 4.2\) for boys, and \(n_2 = 11\) and \(\text{Mean}_2 = 4.6\) for girls. Substitute these values into the formula: \(\text{Overall mean} = \frac{(14 \times 4.2) + (11 \times 4.6)}{25} = 4.38\). This is the overall mean reading level for the class.

Key concepts

Five-Number Summary

The five-number summary is a straightforward way of understanding the distribution of a data set. This summary includes five key values:

• Minimum (smallest value)
• First Quartile (Q1)
• Median (middle value)
• Third Quartile (Q3)
• Maximum (largest value)

In our example, the boys' scores for a reading test are summarized as: 2.0, 3.9, 4.3, 4.9, and 6.0. This means the lowest recorded score among the boys was 2.0 and the highest was 6.0. The median score, which divides the scores into two equal halves, is 4.3. Similarly, for the girls, the scoring records are 2.8, 3.8, 4.5, 5.2, and 5.9. Here, the median, 4.5, indicates the middle reading level among the girls' scores.
These sets of numbers provide a clear picture of score distribution within each group, aiding in direct comparisons and insights into how differently each group performed.

Interquartile Range

The interquartile range (IQR) is a measure of statistical dispersion, which indicates the extent to which the dataset values are spread around the median. To find the IQR, subtract the first quartile (Q1) from the third quartile (Q3).

• For the boys: IQR = Q3 - Q1 = 4.9 - 3.9 = 1.0
• For the girls: IQR = Q3 - Q1 = 5.2 - 3.8 = 1.4

The larger the IQR, the more spread out the scores are around the median. In this scenario, the girls have a larger interquartile range (1.4), suggesting a wider spread of scores in the middle 50% of their data. This could imply greater variability or difference in performance within the girls' scores compared to the boys' scores. Understanding IQR is crucial for assessing consistency and variation in data.

Skewness

Skewness reveals whether the scores are symmetrically distributed around the median, or if they lean more towards one side. Determining skewness involves observing the relative size of two gaps: Q3 to median and median to Q1.

• For the boys: The median is 4.3. The distance between Q3 and the median is 0.6 (4.9 - 4.3), while the distance between the median and Q1 is 0.4 (4.3 - 3.9). This slight asymmetry indicates a mild right skew, meaning there are a few higher scores stretching above the median.
• For the girls: The median is 4.5. Here, both distances (Q3 - median and median - Q1) are 0.7. This balance suggests a symmetric distribution with scores evenly spread around the median.

An understanding of skewness helps identify patterns in data, such as possible outliers or trends, and how well the data aligns with the average (mean) value.

Range Calculation

The range of a dataset is the simplest measure of spread. It tells us how spread out the values are by providing the difference between the maximum and minimum values. Here's how you calculate it:

• For boys: Max - Min = 6.0 - 2.0 = 4.0
• For girls: Max - Min = 5.9 - 2.8 = 3.1

The boys have a greater range of 4.0 compared to the girls' range of 3.1. A larger range means the scores are more spread out. In this case, it indicates that while the boys reached a higher maximum score, their scores also dipped lower than the girls'. Understanding the range gives you a quick overview of the score variability, highlighting the extremes of the data set.