Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/ 84.
The Correct Answer and Explanation is :
To calculate the standard deviation from a frequency distribution, follow these steps:
- Determine the Midpoints: or each class interval, calculate the midpoint by averaging the lower and upper boundaries.
- Compute the Mean: ultiply each midpoint by its corresponding frequency to find the weighted sum. Then, divide this sum by the total number of observations to obtain the mean.
- Calculate Squared Deviations: ubtract the mean from each midpoint to find the deviation, square each deviation, and then multiply by the corresponding frequency.
- Find the Variance: um all the squared deviations and divide by the total number of observations.
- Compute the Standard Deviation: ake the square root of the variance.
Example Calculation:
Consider the following frequency distribution:
| Class Interval | Frequency (f) |
|---|---|
| 10–19 | 5 |
| 20–29 | 8 |
| 30–39 | 12 |
| 40–49 | 10 |
| 50–59 | 5 |
- Midpoints (M): Class Interval Midpoint (M) 10–19 14.5 20–29 24.5 30–39 34.5 40–49 44.5 50–59 54.5
- Mean (μ):
[ \mu = \frac{(14.5 \times 5) + (24.5 \times 8) + (34.5 \times 12) + (44.5 \times 10) + (54.5 \times 5)}{5 + 8 + 12 + 10 + 5} ]
[ \mu = \frac{(72.5) + (196) + (414) + (445) + (272.5)}{40} ]
[ \mu = \frac{1400}{40} = 35 ]
- Squared Deviations (f × (M – μ)²): Class Interval Midpoint (M) Deviation (M – μ) Squared Deviation (M – μ)² f × (M – μ)² 10–19 14.5 -20.5 420.25 2101.25 20–29 24.5 -10.5 110.25 882.00 30–39 34.5 -0.5 0.25 3.00 40–49 44.5 9.5 90.25 902.50 50–59 54.5 19.5 380.25 1901.25
- Variance (σ²):
[ \sigma^2 = \frac{2101.25 + 882.00 + 3.00 + 902.50 + 1901.25}{40} ]
[ \sigma^2 = \frac{5789.00}{40} = 144.725 ]
- Standard Deviation (σ):
[ \sigma = \sqrt{144.725} \approx 12.02 ]
Using the TI-83/84 Calculator:
. Enter Data: – ress [STAT], then select 1:Edit. – nter midpoints into L1 and frequencies into L2.
. Calculate Statistics: – ress [STAT], then select CALC, and choose 1:1-Var Stats. – nput L1, L2 and press [ENTER].
. Interpret Results: – he calculator will display the mean (x̄) and standard deviation (σx for population, Sx for sample).
or a detailed walkthrough, refer to this video:
videoCalculating Standard Deviation on the TI-83 Plus and TI-84 Plusturn0search3