Question

example 3: consider the following dataset: final math grades of 20 grade 7 students (sy 2019 - 2020) grades listed: 98, 90, 91, 84, 87, 83, 96, 79, 85, 82, 92, 81, 86, 85, 88, 80, 89, 88, 80, 76 class interval | tally | frequency 96 - 100 | ii | 2 91 - 95 | ii | 2 86 - 90 | iiii | 5 81 - 85 | iiiii | 5 76 - 80 | iiii | 4 total: 20 step1: r = hs - ls = 92 - 76 = 16 step2: number of interval is - 5 = 16 / 5 = 3

Explanation:

To analyze the given dataset (final math grades of 20 Grade 7 students) and the frequency distribution table, we can check the calculations and the frequency counts:

Step 1: Range Calculation

The formula for the range \( R \) is \( R=\text{HS}-\text{LS} \), where \( \text{HS} \) is the highest score and \( \text{LS} \) is the lowest score.
From the dataset, the highest score (HS) is 98 and the lowest score (LS) is 76.
So, \( R = 98 - 76=22 \) (Note: There seems to be an error in the hand - written calculation where it was written as \( 92 - 76 = 16 \). The correct highest score is 98, not 92).

Step 2: Number of Intervals and Class Width

The number of intervals is given as 5. The class width \( i=\frac{R}{k} \), where \( R \) is the range and \( k \) is the number of intervals.
We have \( R = 22 \) and \( k = 5 \), so \( i=\frac{22}{5}=4.4\approx5 \) (we can round it to a convenient number for class intervals).

Step 3: Checking Frequency Distribution

Let's list out all the scores from the dataset: 98, 90, 91, 84, 87, 83, 96, 79, 85, 82, 92, 81, 86, 85, 88, 80, 89, 88, 80, 76.

• Class interval 96 - 100: Scores in this interval are 98, 96. So frequency = 2 (correct as per the table).
• Class interval 91 - 95: Scores in this interval are 90, 91, 92. Wait, the frequency in the table is 2. But 90, 91, 92 are three scores. There is a mistake here. The correct scores in 91 - 95: 90 (if we consider 90 - 95, but the interval is 91 - 95), 91, 92. So 91 and 92 are in 91 - 95, and 90 is in 90 - 95 (if we adjust the intervals). Maybe the class intervals are defined as 96 - 100, 91 - 95, 86 - 90, 81 - 85, 76 - 80.
• Class interval 86 - 90: Scores: 87, 86, 88, 89, 88. So frequency = 5 (correct as per the table).
• Class interval 81 - 85: Scores: 84, 83, 85, 82, 81, 85. Wait, let's count again: 84, 83, 82, 81, 85, 85. That's 6 scores. But the table says frequency = 4. There is an error here.
• Class interval 76 - 80: Scores: 79, 80, 80, 76. So frequency = 4 (correct as per the table, since 79, 80, 80, 76 are four scores).

To get the correct frequency distribution, we should re - define the class intervals properly and count the frequencies accurately.

If we want to make the frequency distribution correct:

• Let's re - define the class intervals with a class width of 5: 76 - 80, 81 - 85, 86 - 90, 91 - 95, 96 - 100.
• 76 - 80: 76, 79, 80, 80 (frequency = 4)
• 81 - 85: 81, 82, 83, 84, 85, 85 (frequency = 6)
• 86 - 90: 86, 87, 88, 88, 89 (frequency = 5)
• 91 - 95: 90, 91, 92 (frequency = 3)
• 96 - 100: 96, 98 (frequency = 2)

And the sum of frequencies should be equal to the number of students (n = 20). Let's check the sum with the re - calculated frequencies: 4+6 + 5+3+2=20. But in the given table, the sum is 2 + 2+5 + 4+2 = 15 (which is wrong, it should be 20). There are errors in the hand - written frequency distribution table. The correct way is to:

1. Identify the minimum (76) and maximum (98) values.
2. Calculate the range (98 - 76 = 22).
3. Decide on the number of class intervals (let's say 5).
4. Calculate the class width (\(i=\frac{22}{5}\approx5\)).
5. Define the class intervals as 76 - 80, 81 - 85, 86 - 90, 91 - 95, 96 - 100.
6. Count the number of scores in each interval to get the correct frequency.

Answer:

To analyze the given dataset (final math grades of 20 Grade 7 students) and the frequency distribution table, we can check the calculations and the frequency counts:

Step 1: Range Calculation

The formula for the range \( R \) is \( R=\text{HS}-\text{LS} \), where \( \text{HS} \) is the highest score and \( \text{LS} \) is the lowest score.
From the dataset, the highest score (HS) is 98 and the lowest score (LS) is 76.
So, \( R = 98 - 76=22 \) (Note: There seems to be an error in the hand - written calculation where it was written as \( 92 - 76 = 16 \). The correct highest score is 98, not 92).

Step 2: Number of Intervals and Class Width

The number of intervals is given as 5. The class width \( i=\frac{R}{k} \), where \( R \) is the range and \( k \) is the number of intervals.
We have \( R = 22 \) and \( k = 5 \), so \( i=\frac{22}{5}=4.4\approx5 \) (we can round it to a convenient number for class intervals).

Step 3: Checking Frequency Distribution

Let's list out all the scores from the dataset: 98, 90, 91, 84, 87, 83, 96, 79, 85, 82, 92, 81, 86, 85, 88, 80, 89, 88, 80, 76.

• Class interval 96 - 100: Scores in this interval are 98, 96. So frequency = 2 (correct as per the table).
• Class interval 91 - 95: Scores in this interval are 90, 91, 92. Wait, the frequency in the table is 2. But 90, 91, 92 are three scores. There is a mistake here. The correct scores in 91 - 95: 90 (if we consider 90 - 95, but the interval is 91 - 95), 91, 92. So 91 and 92 are in 91 - 95, and 90 is in 90 - 95 (if we adjust the intervals). Maybe the class intervals are defined as 96 - 100, 91 - 95, 86 - 90, 81 - 85, 76 - 80.
• Class interval 86 - 90: Scores: 87, 86, 88, 89, 88. So frequency = 5 (correct as per the table).
• Class interval 81 - 85: Scores: 84, 83, 85, 82, 81, 85. Wait, let's count again: 84, 83, 82, 81, 85, 85. That's 6 scores. But the table says frequency = 4. There is an error here.
• Class interval 76 - 80: Scores: 79, 80, 80, 76. So frequency = 4 (correct as per the table, since 79, 80, 80, 76 are four scores).

To get the correct frequency distribution, we should re - define the class intervals properly and count the frequencies accurately.

If we want to make the frequency distribution correct:

• Let's re - define the class intervals with a class width of 5: 76 - 80, 81 - 85, 86 - 90, 91 - 95, 96 - 100.
• 76 - 80: 76, 79, 80, 80 (frequency = 4)
• 81 - 85: 81, 82, 83, 84, 85, 85 (frequency = 6)
• 86 - 90: 86, 87, 88, 88, 89 (frequency = 5)
• 91 - 95: 90, 91, 92 (frequency = 3)
• 96 - 100: 96, 98 (frequency = 2)

And the sum of frequencies should be equal to the number of students (n = 20). Let's check the sum with the re - calculated frequencies: 4+6 + 5+3+2=20. But in the given table, the sum is 2 + 2+5 + 4+2 = 15 (which is wrong, it should be 20). There are errors in the hand - written frequency distribution table. The correct way is to:

1. Identify the minimum (76) and maximum (98) values.
2. Calculate the range (98 - 76 = 22).
3. Decide on the number of class intervals (let's say 5).
4. Calculate the class width (\(i=\frac{22}{5}\approx5\)).
5. Define the class intervals as 76 - 80, 81 - 85, 86 - 90, 91 - 95, 96 - 100.
6. Count the number of scores in each interval to get the correct frequency.