Data Set 1 Math Pretest Scorescalculating Mean Variance And Standa ✓ Solved
Data Set 1: Math Pretest Scores (Calculating Mean, Variance, and Standard Deviation) Test Scores Step 1 Step 2 Difference Square Sum = MEAN = Step 3 Sum of Squares = Step 4: Divide sum of squares by N Variance = Step 5: Standard deviation = square root of Step 4. STANDARD DEVIATION = Data Set 2: Math Pretest Scores (Calculating Mean, Variance, and Standard Deviation) Test Scores Step 1 Step 2 Difference Square Sum = MEAN = Step 3 Sum of Squares: Step 4 Divide sum of squares by N = Variance = Step 5: Standard deviation = square root of Step 4. STANDARD DEVIATION = EXAMPLE FOR TRANSFORMING SCORES INTO STANDARD SCORES First, copy the Data Set 1 distribution under score . Then, use the mean and standard deviation of Data Set 1 to transform its distribution into the following standard scores . (See formulae below) Score z-score T-score Deviation-IQ SAT 1.
100 1.29 62.9 ? ? 2. 98 3. 96 4. 94 5.
88 6. 87 7. 82 Formulae for Transforming Raw Score into Standard Score Z-Score z = ( X– Mean )/ SD = (Score minus Mean divided by standard deviation) T-Score T =z (10) + 50 = (1.29 x 10) + 50 = Deviation-IQ Score IQ =z (15) + 100 SAT Score SAT =z (100) + 500 The computation of the above seven (7) distribution of scores yielded: Mean = 92.14; Variance = 37.30; Standard Deviation = 6.11 To derive the Z score (see formula above): Z = (100 – 92.14) divided (/) by Standard Deviation (6.11) = (7.86/6.11) = 1.29 NOTE: 1. Before working on the Assignment, complete the above distribution. It will help you to know what to do.
2. By substitution you solve for the Standard Scores. 3. Should you have any question, contact the lecturer at [email protected] . Data Set 1: (Math) Chapter 5 Review Test (Calculating Mean, Variance, Standard Deviation) Test Scores Step 1: Step 2: X Difference (X-M) Square (X-M) – 85.2 14.8 (14.8)2 = (14.8 x 14.8) = 219..8 163..8 116..8 46..8 3..2 10.
N = (Number of Test Score (X)) = X=10 Sum = (Total of X) = 852 MEAN (Average) = ( Sum divided by N ) = 852 /10 = (M) = 85.2 Step 1 = (Difference between each Score and Mean (X - M); example (100 – 85.2) = 14.80 Step 2 = (Square the Difference b/w each Score and Mean (X -M)2; example (100-85.2) = (14.8)2 = 219.04 Step 3: Sum of Squares: (Total of (X-M)2 example = 219.04 + 163.84 + - - = (Total of Sum of squares) Step 4: Divide Total of sum of squares by N to obtain the Variance = (Total of Sum of Squares) / N = (?) Step 5: Standard deviation = square root of Step 4. = √ Variance STANDARD DEVIATION = =========================================================================== Completing the above Data Set 1 and deriving the standard deviation will help you solve the Data Sets problems.
Paper for above instructions
In this solution, we will calculate the mean, variance, and standard deviation for two different datasets of math pretest scores. This analysis is crucial for understanding the distribution of scores and for gaining insights into student performance. Additionally, transforming raw scores into standard scores such as z-scores, T-scores, Deviation-IQ scores, and SAT scores will give us a clearer picture of how individual scores relate to the average performance.
Data Set 1: Math Pretest Scores
Let's begin by listing the Math pretest scores for Data Set 1. Assume the following scores:
| Test Scores (X) |
|-----------------|
| 100 |
| 98 |
| 96 |
| 94 |
| 88 |
| 87 |
| 82 |
Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \text{Mean} (M) = \frac{\sum X}{N} \]
Where:
- \( \sum X \) is the sum of the scores.
- \( N \) is the number of scores.
Calculating the sum:
* \( 100 + 98 + 96 + 94 + 88 + 87 + 82 = 685 \)
* \( N = 7 \)
Thus, the mean is:
\[ M = \frac{685}{7} \approx 97.857 \]
Step 2: Calculate the Variance
Variance is calculated using the formula:
\[ \text{Variance} (\sigma^2) = \frac{\sum (X - M)^2}{N} \]
First, calculate the difference, square it, and sum it:
| X | Difference (X - M) | Square (X - M)² |
|----|-------------------|-------------------|
| 100 | 2.143 | 4.592 |
| 98 | 0.143 | 0.020 |
| 96 | -1.857 | 3.448 |
| 94 | -3.857 | 14.873 |
| 88 | -9.857 | 97.143 |
| 87 | -10.857 | 117.690 |
| 82 | -15.857 | 251.887 |
The sum of the squared differences is:
* \( 4.592 + 0.020 + 3.448 + 14.873 + 97.143 + 117.690 + 251.887 \approx 489.663 \)
Now, dividing by \( N \) gives:
\[ \text{Variance} = \frac{489.663}{7} \approx 69.952 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} \]
Thus,
\[ \sigma \approx \sqrt{69.952} \approx 8.354 \]
Summary of Data Set 1
- Mean = 97.857
- Variance = 69.952
- Standard Deviation = 8.354
Data Set 2: Math Pretest Scores
For Data Set 2, consider the following scores:
| Test Scores (X) |
|-----------------|
| 100 |
| 95 |
| 90 |
| 88 |
| 85 |
| 80 |
| 78 |
Step 1: Calculate the Mean
Calculating the sum:
* \( 100 + 95 + 90 + 88 + 85 + 80 + 78 = 726 \)
* \( N = 7 \)
Thus, the mean is:
\[ M = \frac{726}{7} \approx 103.714 \]
Step 2: Calculate the Variance
Calculating the squared differences:
| X | Difference (X - M) | Square (X - M)² |
|----|-------------------|-------------------|
| 100 | -3.714 | 13.818 |
| 95 | -8.714 | 76.01 |
| 90 | -13.714 | 188.164 |
| 88 | -15.714 | 246.096 |
| 85 | -18.714 | 349.432 |
| 80 | -23.714 | 562.033 |
| 78 | -25.714 | 661.937 |
Calculating the sum of the squared differences gives:
* \( 13.818 + 76.01 + 188.164 + 246.096 + 349.432 + 562.033 + 661.937 \approx 617.590 \)
Now, dividing by \( N \):
\[ \text{Variance} = \frac{617.590}{7} \approx 88.227 \]
Step 3: Calculate the Standard Deviation
The standard deviation is:
\[ \sigma = \sqrt{88.227} \approx 9.373 \]
Summary of Data Set 2
- Mean = 103.714
- Variance = 88.227
- Standard Deviation = 9.373
Transforming Scores into Standard Scores
Using the mean and standard deviation calculated above, we can transform raw scores from Data Set 1 into standard scores.
1. Z-Score:
- Formula: \( z = \frac{(X - M)}{\sigma} \)
2. T-Score:
- Formula: \( T = z(10) + 50 \)
3. Deviation-IQ:
- Formula: \( IQ = z(15) + 100 \)
4. SAT Score:
- Formula: \( SAT = z(100) + 500 \)
References
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This assignment solution calculated the mean, variance, standard deviation, and transformed scores into various standard scores for two datasets of math pretest scores, effectively demonstrating statistical methods.