Let's solve the problem step-by-step.

Given Data:

The number of detergent bars sold daily by a grocery store over 10 days: $34, 22, 23, 27, 16, 35, 25, 18, 33, 37$

We need to find the following:

1. Mean (average)
2. Median (middle value)
3. Mode (most frequent value)
4. MAD (Mean Absolute Deviation)
5. Variance
6. Standard Deviation

1. Mean:

The mean is calculated by summing all the values and dividing by the number of observations.

$\text{Mean} = \frac{\sum \text{(All Values)}}{\text{Number of Values}} = \frac{34 + 22 + 23 + 27 + 16 + 35 + 25 + 18 + 33 + 37}{10}$

$\text{Mean} = \frac{270}{10} = 27$

2. Median:

To find the median, we need to arrange the data in ascending order:

$16, 18, 22, 23, 25, 27, 33, 34, 35, 37$

Since there are 10 values (even number), the median will be the average of the 5th and 6th values:

$\text{Median} = \frac{25 + 27}{2} = 26$

3. Mode:

The mode is the number that appears most frequently. In this dataset, all numbers appear only once, so there is no mode.

4. Mean Absolute Deviation (MAD):

MAD is calculated as the average of the absolute deviations from the mean.

First, calculate the absolute deviations:

$|34 - 27| = 7, \; |22 - 27| = 5, \; |23 - 27| = 4, \; |27 - 27| = 0, \; |16 - 27| = 11$ $|35 - 27| = 8, \; |25 - 27| = 2, \; |18 - 27| = 9, \; |33 - 27| = 6, \; |37 - 27| = 10$

Now, find the MAD:

$\text{MAD} = \frac{7 + 5 + 4 + 0 + 11 + 8 + 2 + 9 + 6 + 10}{10} = \frac{62}{10} = 6.2$

5. Variance:

Variance is calculated as the average of the squared deviations from the mean.

First, calculate the squared deviations:

$(34 - 27)^2 = 49, \; (22 - 27)^2 = 25, \; (23 - 27)^2 = 16, \; (27 - 27)^2 = 0, \; (16 - 27)^2 = 121$ $(35 - 27)^2 = 64, \; (25 - 27)^2 = 4, \; (18 - 27)^2 = 81, \; (33 - 27)^2 = 36, \; (37 - 27)^2 = 100$

Now, find the variance:

$\text{Variance} = \frac{49 + 25 + 16 + 0 + 121 + 64 + 4 + 81 + 36 + 100}{10} = \frac{496}{10} = 49.6$

6. Standard Deviation:

The standard deviation is the square root of the variance:

$\text{Standard Deviation} = \sqrt{49.6} \approx 7.04$

Final Answers:

1. Mean: 27
2. Median: 26
3. Mode: None
4. MAD: 6.2
5. Variance: 49.6
6. Standard Deviation: 7.04

Would you like more details on any part of this solution or have any questions?

Related Questions:

1. How do you find the quartiles for the given dataset?
2. What is the range of this data, and how is it calculated?
3. Can you explain the significance of variance and standard deviation in data analysis?
4. What would the data look like if there was a mode?
5. How would the calculations change if an additional day of data was included?

Tip: Understanding the difference between variance and standard deviation can help in interpreting the spread of data more accurately.