Question

using technology, find the range and standard deviation. round to two decimals places.
the table below gives the number of hours spent watching tv last week by a sample of 24 children.
27 84 82 73 87 31
51 59 58 74 27 55
84 44 35 54 22 32
27 38 73 26 99 15
range =
standard deviation =

Explanation:

Step1: Find the range

The range is calculated as the maximum value minus the minimum value. First, we identify the maximum and minimum values from the data set.
Looking at the data: 27, 84, 82, 73, 87, 31, 51, 59, 58, 74, 27, 55, 84, 44, 35, 54, 22, 32, 27, 38, 73, 26, 99, 15.
The maximum value is 99 and the minimum value is 15.
So, Range = Maximum - Minimum = 99 - 15 = 84.

Step2: Find the standard deviation (sample standard deviation)

To find the standard deviation, we can use the formula for the sample standard deviation: $s = \sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}}$, where $x_i$ are the data points, $\bar{x}$ is the sample mean, and $n$ is the sample size.
First, calculate the sample mean $\bar{x}$.
The sum of the data points:
27 + 84 + 82 + 73 + 87 + 31 + 51 + 59 + 58 + 74 + 27 + 55 + 84 + 44 + 35 + 54 + 22 + 32 + 27 + 38 + 73 + 26 + 99 + 15.
Let's calculate the sum:
27*3 = 81; 84*2 = 168; 82 = 82; 73*2 = 146; 87 = 87; 31 = 31; 51 = 51; 59 = 59; 58 = 58; 74 = 74; 55 = 55; 44 = 44; 35 = 35; 54 = 54; 22 = 22; 32 = 32; 38 = 38; 26 = 26; 99 = 99; 15 = 15.
Now sum all these:
81 + 168 = 249; 249 + 82 = 331; 331 + 146 = 477; 477 + 87 = 564; 564 + 31 = 595; 595 + 51 = 646; 646 + 59 = 705; 705 + 58 = 763; 763 + 74 = 837; 837 + 81 (wait, no, 27*3 is 81, then next is 55: 837 + 55 = 892; 892 + 168 (84*2) = 1060; 1060 + 44 = 1104; 1104 + 35 = 1139; 1139 + 54 = 1193; 1193 + 22 = 1215; 1215 + 32 = 1247; 1247 + 38 = 1285; 1285 + 26 = 1311; 1311 + 99 = 1410; 1410 + 15 = 1425. Wait, that can't be right. Let's add all numbers one by one:
27, 84 (27+84=111), 82 (111+82=193), 73 (193+73=266), 87 (266+87=353), 31 (353+31=384), 51 (384+51=435), 59 (435+59=494), 58 (494+58=552), 74 (552+74=626), 27 (626+27=653), 55 (653+55=708), 84 (708+84=792), 44 (792+44=836), 35 (836+35=871), 54 (871+54=925), 22 (925+22=947), 32 (947+32=979), 27 (979+27=1006), 38 (1006+38=1044), 73 (1044+73=1117), 26 (1117+26=1143), 99 (1143+99=1242), 15 (1242+15=1257). Ah, I made a mistake earlier. So the sum is 1257.
The sample size $n = 24$.
So the sample mean $\bar{x} = \frac{1257}{24} = 52.375$.

Next, calculate the sum of squared deviations: $\sum_{i = 1}^{n}(x_i - \bar{x})^2$.
We calculate each $(x_i - \bar{x})^2$:
For 27: $(27 - 52.375)^2 = (-25.375)^2 = 643.890625$ (and there are 3 such values: 27,27,27, so 3*643.890625 = 1931.671875)
For 84: $(84 - 52.375)^2 = (31.625)^2 = 999.140625$ (two values: 84,84, so 2*999.140625 = 1998.28125)
For 82: $(82 - 52.375)^2 = (29.625)^2 = 877.640625$ (one value)
For 73: $(73 - 52.375)^2 = (20.625)^2 = 425.390625$ (two values: 73,73, so 2*425.390625 = 850.78125)
For 87: $(87 - 52.375)^2 = (34.625)^2 = 1198.921875$ (one value)
For 31: $(31 - 52.375)^2 = (-21.375)^2 = 456.890625$ (one value)
For 51: $(51 - 52.375)^2 = (-1.375)^2 = 1.890625$ (one value)
For 59: $(59 - 52.375)^2 = (6.625)^2 = 43.890625$ (one value)
For 58: $(58 - 52.375)^2 = (5.625)^2 = 31.640625$ (one value)
For 74: $(74 - 52.375)^2 = (21.625)^2 = 467.640625$ (one value)
For 55: $(55 - 52.375)^2 = (2.625)^2 = 6.890625$ (one value)
For 44: $(44 - 52.375)^2 = (-8.375)^2 = 70.140625$ (one value)
For 35: $(35 - 52.375)^2 = (-17.375)^2 = 301.890625$ (one value)
For 54: $(54 - 52.375)^2 = (1.625)^2 = 2.640625$ (one value)
For 22: $(22 - 52.375)^2 = (-30.375)^2 = 922.640625$ (one value)
For 32: $(32 - 52.375)^2 = (-20.375)^2 = 415.140625$ (one value)
For 38: $(38 - 52.375)^2 = (-14.375)^2 = 206.640625$ (one value)
For 26: $(26 - 52.375)^2 = (-26.375)^2 = 695.640625$ (one value)
For 99: $(99 - 52.375)^2 = (46.625)^2 = 2174.015625$ (one value)
For 15: $(15 - 52.375)^2 = (-37.375)^2 = 1397.015625$ (one value)

Now sum all these squared deviations:
1931.671875 (from 27s) + 1998.28125 (from 84s) + 877.640625 (82) + 850.78125 (73s) + 1198.921875 (87) + 456.890625 (31) + 1.890625 (51) + 43.890625 (59) + 31.640625 (58) + 467.640625 (74) + 6.890625 (55) + 70.140625 (44) + 301.890625 (35) + 2.640625 (54) + 922.640625 (22) + 415.140625 (32) + 206.640625 (38) + 695.640625 (26) + 2174.015625 (99) + 1397.015625 (15)

Let's calculate step by step:
1931.671875 + 1998.28125 = 3929.953125
3929.953125 + 877.640625 = 4807.59375
4807.59375 + 850.78125 = 5658.375
5658.375 + 1198.921875 = 6857.296875
6857.296875 + 456.890625 = 7314.1875
7314.1875 + 1.890625 = 7316.078125
7316.078125 + 43.890625 = 7359.96875
7359.96875 + 31.640625 = 7391.609375
7391.609375 + 467.640625 = 7859.25
7859.25 + 6.890625 = 7866.140625
7866.140625 + 70.140625 = 7936.28125
7936.28125 + 301.890625 = 8238.171875
8238.171875 + 2.640625 = 8240.8125
8240.8125 + 922.640625 = 9163.453125
9163.453125 + 415.140625 = 9578.59375
9578.59375 + 206.640625 = 9785.234375
9785.234375 + 695.640625 = 10480.875
10480.875 + 2174.015625 = 12654.890625
12654.890625 + 1397.015625 = 14051.90625

Now, the sample standard deviation $s = \sqrt{\frac{14051.90625}{24 - 1}} = \sqrt{\frac{14051.90625}{23}}$.
Calculate $\frac{14051.90625}{23} \approx 610.952445652$.
Then $s = \sqrt{610.952445652} \approx 24.72$.

Answer:

Range = 84.00
Standard Deviation = 24.72