Calculate the sample standard deviation and sample variance for the following frequency distribution of final exam scores in a statistics class. If necessary, round to one more decimal place than the largest number of decimal places given in the data.
| Class | Frequency |
|---|---|
| 41 - 5241 - 52 | 1111 |
| 53 - 6453 - 64 | 66 |
| 65 - 7665 - 76 | 77 |
| 77 - 8877 - 88 | 1414 |
| 89 - 10089 - 100 | 99 |
| Class | Frequency(f) | Midpoint (m) | f*m | (m - x̅) | (m - x̅)² | f* (m - x̅)² |
| 41 - 52 | 11 | 46.5 | 511.5 | -25.0212766 | 626.0642827 | 6886.7071 |
| 53 - 64 | 6 | 58.5 | 351 | -13.0212766 | 169.5536443 | 1017.3219 |
| 65 - 76 | 7 | 70.5 | 493.5 | -1.0212766 | 1.043005894 | 7.3010413 |
| 77 - 88 | 14 | 82.5 | 1155 | 10.9787234 | 120.5323675 | 1687.4531 |
| 89 - 100 | 9 | 94.5 | 850.5 | 22.9787234 | 528.0217291 | 4752.1956 |
| Total= | 47 | 3361.5 | 14350.979 |
| Mean (x̅) =∑(F*m)/ ∑f =3361.5/47 =71.5212766 |
| Standard Deviation (s) = √(∑( f* (m - x̅)²)/ ∑f-1 ) = √(14350.979/46) =17.66289343 |
| Variance (s²) = (Standard Deviation ) ² =(17.66289343) ² = 311.9778043 |
Sample Standard Deviation = 17.7
Sample Variance = 312
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