find the mean, median, mode and range for each set of data. calculator usage is encouraged! 1. 23, 87, 19…

1. For the data - set (23,87,19,34,37,87,81,5,14,100,26)

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 11$ and $\sum_{i=1}^{11}x_{i}=23 + 87+19+34+37+87+81+5+14+100+26=513$. So, $\bar{x}=\frac{513}{11}\approx46.64$.

Step2: Find the median

First, arrange the data in ascending order: (5,14,19,23,26,34,37,81,87,87,100). Since (n = 11) (odd), the median is the (\left(\frac{n + 1}{2}\right))-th value. So, the median is the 6 - th value, which is (34).

Step3: Determine the mode

The mode is the value that appears most frequently. Here, (87) appears twice and other values appear once, so the mode is (87).

Step4: Calculate the range

The range is the difference between the maximum and minimum values. Range (=100 - 5=95).

2. For the data - set (10,11,17,14,2,13,26)

Explanation:

Step1: Calculate the mean

$n = 7$, $\sum_{i=1}^{7}x_{i}=10 + 11+17+14+2+13+26=93$. So, $\bar{x}=\frac{93}{7}\approx13.29$.

Step2: Find the median

Arrange the data in ascending order: (2,10,11,13,14,17,26). Since (n = 7) (odd), the median is the (\left(\frac{n+1}{2}\right))-th value, which is the 4 - th value, so the median is (13).

Step3: Determine the mode

Each value appears only once, so there is no mode.

Step4: Calculate the range

Range (=26 - 2=24).

3. For the data - set (86,86,24,92,37,14,52,24,63)

Explanation:

Step1: Calculate the mean

$n = 9$, $\sum_{i=1}^{9}x_{i}=86+86+24+92+37+14+52+24+63=478$. So, $\bar{x}=\frac{478}{9}\approx53.11$.

Step2: Find the median

Arrange the data in ascending order: (14,24,24,37,52,63,86,86,92). Since (n = 9) (odd), the median is the (\left(\frac{n + 1}{2}\right))-th value, which is the 5 - th value, so the median is (52).

Step3: Determine the mode

Both (24) and (86) appear twice, so the modes are (24) and (86).

Step4: Calculate the range

Range (=92 - 14=78).

4. For the data - set (27,54,6,31,57,22,19,12)

Explanation:

Step1: Calculate the mean

$n = 8$, $\sum_{i=1}^{8}x_{i}=27+54+6+31+57+22+19+12=228$. So, $\bar{x}=\frac{228}{8}=28.5$.

Step2: Find the median

Arrange the data in ascending order: (6,12,19,22,27,31,54,57). Since (n = 8) (even), the median is the average of the (\frac{n}{2})-th and (\left(\frac{n}{2}+1\right))-th values. The 4 - th and 5 - th values are (22) and (27), so the median is $\frac{22 + 27}{2}=24.5$.

Step3: Determine the mode

Each value appears only once, so there is no mode.

Step4: Calculate the range

Range (=57 - 6=51).

5. For the data - set (8,32,41,6,36,10,37,31,29,37,32,11,30,15,23,37,34,17,29)

Explanation:

Step1: Calculate the mean

$n = 19$, $\sum_{i=1}^{19}x_{i}=8+32+41+6+36+10+37+31+29+37+32+11+30+15+23+37+34+17+29=465$. So, $\bar{x}=\frac{465}{19}\approx24.47$.

Step2: Find the median

Arrange the data in ascending order: (6,8,10,11,15,17,23,29,29,30,31,32,32,34,36,37,37,37,41). Since (n = 19) (odd), the median is the (\left(\frac{n + 1}{2}\right))-th value, which is the 10 - th value, so the median is (30).

Step3: Determine the mode

The value (37) appears 3 times, more frequently than other values, so the mode is (37).

Step4: Calculate the range

Range (=41 - 6=35).

6. For the data - set (9.3,1.6,8.2,4.7,8.9,3,11.4,2.1,1.6)

Explanation:

Step1: Calculate the mean

$n = 9$, $\sum_{i=1}^{9}x_{i}=9.3+1.6+8.2+4.7+8.9+3+11.4+2.1+1.6=50.8$. So, $\bar{x}=\frac{50.8}{9}\approx5.64$.

Step2: Find the median

Arrange the data in ascending order: (1.6,1.6,2.1,3,4.7,8.2,8.9,9.3,11.4). Since (n = 9) (odd), the median is the (\left(\frac{n + 1}{2}\right))-th value, which is the 5 - th value, so the median is (4.7).

Step3: Determine the mode

The value (1.6) appears twice, so the mode is (1.6).

Step4: Calculate the range

Range (=11.4 - 1.6=9.8).

Answer:

1. Mean: $\approx46.64$, Median: (34), Mode: (87), Range: (95)
2. Mean: $\approx13.29$, Median: (13), Mode: None, Range: (24)
3. Mean: $\approx53.11$, Median: (52), Modes: (24,86), Range: (78)
4. Mean: (28.5), Median: (24.5), Mode: None, Range: (51)
5. Mean: $\approx24.47$, Median: (30), Mode: (37), Range: (35)
6. Mean: $\approx5.64$, Median: (4.7), Mode: (1.6), Range: (9.8)