what is the mean and the standard deviation of the data in the table given? round your answer to 2 decimal places
value: 4.8, 5.2, 5.4, 5.9, 6.5, 6.6
frequency: 25, 21, 23, 17, 24, 25
Boo boo goo goo ha ha
To find the mean and standard deviation of the data given, we need to calculate the weighted mean and weighted standard deviation.
Step 1: Calculate the product of each value and its corresponding frequency.
- For the value 4.8, its frequency is 25, so the product is 4.8 * 25 = 120.
- For the value 5.2, its frequency is 21, so the product is 5.2 * 21 = 109.2.
- For the value 5.4, its frequency is 23, so the product is 5.4 * 23 = 124.2.
- For the value 5.9, its frequency is 17, so the product is 5.9 * 17 = 100.3.
- For the value 6.5, its frequency is 24, so the product is 6.5 * 24 = 156.
- For the value 6.6, its frequency is 25, so the product is 6.6 * 25 = 165.
Step 2: Sum up all the products calculated in Step 1.
120 + 109.2 + 124.2 + 100.3 + 156 + 165 = 774.7.
Step 3: Sum up all the frequencies.
25 + 21 + 23 + 17 + 24 + 25 = 135.
Step 4: Calculate the weighted mean by dividing the sum of the products by the sum of the frequencies.
Mean = 774.7 / 135 ≈ 5.73 (rounded to 2 decimal places).
Step 5: Calculate the standard deviation.
a. Calculate the squared difference between each value and the mean, multiplied by the corresponding frequency.
- For the value 4.8, the squared difference is (4.8 - 5.73)² * 25 = 4.40.
- For the value 5.2, the squared difference is (5.2 - 5.73)² * 21 = 4.57.
- For the value 5.4, the squared difference is (5.4 - 5.73)² * 23 = 2.81.
- For the value 5.9, the squared difference is (5.9 - 5.73)² * 17 = 0.66.
- For the value 6.5, the squared difference is (6.5 - 5.73)² * 24 = 13.96.
- For the value 6.6, the squared difference is (6.6 - 5.73)² * 25 = 20.13.
b. Sum up all the squared differences calculated in Step 5a.
4.40 + 4.57 + 2.81 + 0.66 + 13.96 + 20.13 = 46.53.
c. Calculate the variance by dividing the sum of squared differences by the sum of frequencies.
Variance = 46.53 / 135 ≈ 0.3454 (rounded to 4 decimal places).
d. Calculate the standard deviation by taking the square root of the variance.
Standard Deviation ≈ √0.3454 ≈ 0.5881 (rounded to 4 decimal places) or 0.59 (rounded to 2 decimal places).
Therefore, the mean is approximately 5.73 and the standard deviation is approximately 0.59 (rounded to 2 decimal places).