Use the following data (first exam scores) from Susan Dean's spring pre-calculus class: 
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100
Calculate the following: 
i. The mean
ii. The mode
iii. The median 
iv. The first quartile
v. The third quartile 
vi. InterQuartile Range

Let's analyze the given data set of first exam scores from Susan Dean's spring pre-calculus class:

Data Set (sorted):  
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100

### i. The Mean
The mean is the average of the data set, calculated by summing all values and dividing by the number of values.

\[
\text{Mean} = \frac{\sum \text{scores}}{n} 
\]
where \( n \) is the number of scores.

\[
\text{Mean} = \frac{33 + 42 + 49 + 49 + 53 + 55 + 55 + 61 + 63 + 67 + 68 + 68 + 69 + 69 + 72 + 73 + 74 + 78 + 80 + 83 + 88 + 88 + 88 + 90 + 92 + 94 + 94 + 94 + 94 + 96 + 100}{31}
\]

Calculating the sum:

\[
\sum \text{scores} = 2161
\]

So,

\[
\text{Mean} = \frac{2161}{31} \approx 69.71
\]

### ii. The Mode
The mode is the value(s) that appear most frequently in the data set.

Looking at the data, the mode is:

\[
\text{Mode} = 94 \quad (\text{appears 4 times})
\]

### iii. The Median
The median is the middle value when the data is ordered from least to greatest. Since there are 31 scores, the median will be the 16th value.

\[
\text{Median} = 73
\]

### iv. The First Quartile (Q1)
The first quartile (Q1) is the median of the lower half of the data (not including the overall median if the number of data points is odd). The lower half of the data consists of the first 15 values:

33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72

The median of this subset is the 8th value:

\[
Q1 = 61
\]

### v. The Third Quartile (Q3)
The third quartile (Q3) is the median of the upper half of the data. The upper half of the data consists of the last 15 values:

73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100

The median of this subset is the 8th value:

\[
Q3 = 88
\]

### vi. Interquartile Range (IQR)
The interquartile range (IQR) is the difference between the third quartile and the first quartile:

\[
\text{IQR} = Q3 - Q1 = 88 - 61 = 27
\]

### Summary:
- **Mean**: 69.71
- **Mode**: 94
- **Median**: 73
- **First Quartile (Q1)**: 61
- **Third Quartile (Q3)**: 88
- **Interquartile Range (IQR)**: 27

Would you like more details or have any questions about these calculations? Here are some related questions to explore further:

1. How would removing the outlier (if any) affect the mean?
2. What is the range of the data?
3. How would you calculate the variance and standard deviation for this data set?
4. How does the skewness of this data set affect the mean and median?
5. How can box plots be used to visually represent the quartiles and IQR?

**Tip:** When analyzing a data set, it's helpful to look at the shape of the distribution (skewed, symmetric, etc.) to understand how the mean and median might differ.

Explore the analysis of Susan Dean's spring pre-calculus class first exam scores. Calculate the mean, mode, median, first quartile, third quartile, and interquartile range from the dataset provided. Learn about descriptive statistics and their application in educational assessments.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation