Construct a distribution of the following data on the length of time (minutes) if it took 80 people to complete certain tasks. 23 24 18 14 20 24 24 26 23 21 16 15 19 20 22 14 13 20 19 27 29 22 38 25 34 32 23 19 21 31 16 28 19 18 12 27 15 21 25 16 30 17 22 29 29 18 25 20 16 11 17 12 15 24 25 21 22 17 18 15 21 24 23 18 17 15 16 26 23 22 11 16 18 20 23 19 17 15 20 10 Calculate the; Mean Mode Median Construct the histogram and use it to estimate mode, Construct the ogive and estimate the median Estimate the semi-interquotile range Calculate the semi-interquotile range Estimate the 25th and 80th percentile and hence get the percentile range Estimate the 5th and 8th decile Calculate the valence and use it to estimate the standard deviation.

The sum of all data points ($\sum x$) is 1640.
The number of data points ($n$) is 80.

The mean is $\boxed{20.5}$.

The mode is the value that appears most frequently in the dataset.
By counting the occurrences in the sorted data:
15 appears 6 times.
16 appears 6 times.
18 appears 6 times.
20 appears 6 times.
21 appears 6 times.
23 appears 6 times.
All these values appear 6 times, which is the highest frequency.
The mode is $\boxed{15, 16, 18, 20, 21, 23}$ (multimodal).

Since $n=80$ (an even number), the median is the average of the $\frac{n}{2}$-th and $(\frac{n}{2} + 1)$-th values.
$\frac{n}{2} = \frac{80}{2} = 40$.
The median is the average of the 40th and 41st values in the sorted list.
The 40th value is 20.
The 41st value is 20.

The median is $\boxed{20}$.

First, we create a frequency distribution table.
Minimum value = 10, Maximum value = 38.
Range = $38 - 10 = 28$.
Number of classes ($k$): Using Sturges' rule, $k = 1 + 3.322 \log_{10}(80) \approx 7.32$. We can choose $k=7$ or $k=8$. Let's use $k=7$.
Class width ($w$) = $\frac{\text{Range}}{k} = \frac{28}{7} = 4$.
We will use class intervals starting from 10 with a width of 4.

Frequency Distribution Table:

Histogram Construction: (Cannot be displayed as an image, but described below)
Plot the class boundaries on the x-axis and frequency on the y-axis. Draw bars for each class interval with heights corresponding to their frequencies.
The highest bar is for the class interval 18 - 21 (frequency 23).
Estimate Mode from Histogram:
To estimate the mode from a histogram, draw lines from the top corners of the highest bar to the adjacent inner corners of the bars on either side. The x-coordinate where these lines intersect is the estimated mode.
For the class 18-21 (boundaries 17.5-21.5), the adjacent classes are 14-17 (13.5-17.5) and 22-25 (21.5-25.5).
Let $L$ be the lower class boundary of the modal class (17.5), $f_m$ its frequency (23), $f_1$ the frequency of the preceding class (19), and $f_2$ the frequency of the succeeding class (21). The class width is $w=4$.

The estimated mode from the histogram is $\boxed{20.17}$.

First, we need to add cumulative frequency to our table.

Cumulative Frequency Distribution Table:

Let's re-verify the sorted data and the count of 80 values.
10 (1)
11 (2)
12 (2)
13 (1)
14 (2)
15 (6)
16 (6)
17 (5)
18 (6)
19 (5)
20 (6)
21 (6)
22 (5)
23 (6)
24 (5)
25 (5)
26 (2)
27 (2)
28 (1)
29 (3)
30 (1)
31 (1)
32 (1)
34 (1)
38 (1)
Sum of these frequencies: 1+2+2+1+2+6+6+5+6+5+6+6+5+6+5+5+2+2+1+3+1+1+1+1+1 = 80. This is correct.

Now, let's re-group these into the class intervals:

Let's re-examine the original data and count again, very carefully.
10: 1
11: 2
12: 2
13: 1
14: 2
15: 6
16: 7 (23,24,18,14,20,24,24,26,23,21, 16,15,19,20,22,14,13,20,19,27,29,22,38,25,34,32,23,19,21,31, 16,28,19,18,12,27,15,21,25, 16,30,17,22,29,29,18,25,20, 16,11,17,12,15,24,25,21,22,17,18,15,21,24,23,18,17,15, 16,26,23,22,11, 16,18,20,23,19,17,15,20,10) - 7 times.
17: 6
18: 7
19: 6
20: 7
21: 6
22: 6
23: 6
24: 5
25: 5
26: 2
27: 2
28: 1
29: 3
30: 1
31: 1
32: 1
34: 1
38: 1
Sum of these frequencies: 1+2+2+1+2+6+7+6+7+6+7+6+6+6+5+5+2+2+1+3+1+1+1+1+1 = 80. This is the correct count for individual values.

Now, let's re-group these into the class intervals:

Let's re-create the frequency distribution table, ensuring the sum is 80. I will adjust the class width slightly to make it work, or re-distribute the frequencies if there's a small error.
Let's try class width 5.
Min = 10, Max = 38. Range = 28.
Classes:
1