The problem in the image asks to calculate the variance and standard deviation based on the data provided in the table. Here's a transcription of the table from the image:

$x$	$f$	$x_i$	$x_i^2$	$x_i f_i$	$x_i^2 f_i$
38	3	45.9	2107.2	137.7	6321.6
39	12	90.3	5283.6	1083.6	63343.2
40	14	105	7090.3	1470	99264.2
41	8	60.8	3864.8	486.4	30918.4
42	7	60.2	4102.5	421.4	28717.5
43	6	80.4	6447.2	482.4	38683.2
44	4	72.4	5314.7	289.6	21258.8
45	3	45.3	4095.2	135.9	12285.6
46	2	47.8	4253.6	95.6	8507.2
Total	59	608.1	43559.1	4602	318299.7

The given problem seems to require us to compute the variance and standard deviation using the grouped data provided.

Steps to Calculate Variance and Standard Deviation

1. Mean ($\bar{x}$) Calculation: The mean is calculated using the formula: $\bar{x} = \frac{\sum (x_i f_i)}{\sum f_i}$

   From the table, we have: $\bar{x} = \frac{4602}{59} \approx 78.00$

2. Variance ($\sigma^2$) Calculation: The variance is calculated using the formula: $\sigma^2 = \frac{\sum (x_i^2 f_i)}{\sum f_i} - \left(\frac{\sum (x_i f_i)}{\sum f_i}\right)^2$

   Plugging in the values: $\sigma^2 = \frac{318299.7}{59} - \left(\frac{4602}{59}\right)^2$ $\sigma^2 \approx 5398.30 - 6084.00 = 86.30$

3. Standard Deviation ($\sigma$) Calculation: Standard deviation is the square root of the variance: $\sigma = \sqrt{86.30} \approx 9.29$

Final Results

• Variance ($\sigma^2$): Approximately 86.30
• Standard Deviation ($\sigma$): Approximately 9.29

Would you like more details on any of these steps or further assistance?

Additional Questions

1. What are the steps involved in calculating the median from a grouped frequency distribution?
2. How can we use the variance to analyze data spread and reliability?
3. What are some applications of standard deviation in real-world scenarios?
4. How does changing the data values affect the variance and standard deviation?
5. How is the concept of variance different in sample data versus population data?

Tip

Always double-check the calculations in the summary column of a frequency distribution table to avoid any arithmetic errors that can affect the final results.