Question

the following data give the test scores for all 13 students in a small calculus class. 100 61 73 76 87 84 81 76 80 61 63 66 61. what is the mean of this data set? 72.99 70.39 74.54 70.21. what is the mode of this data set? 80 61 76 73. what is the variance of this data set? 118.4187 132.8639 115.8094 143.9359. what is the standard deviation of this data set? 10.882 11.5267

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 13$ and $\sum_{i=1}^{13}x_i=100 + 61+73+76+87+84+81+76+80+61+63+66+61=969$. So, $\bar{x}=\frac{969}{13}\approx74.54$.

Step2: Find the mode

The mode is the number that appears most frequently in the data - set. The number 61 appears 3 times, more frequently than any other number in the set $\{100,61,73,76,87,84,81,76,80,61,63,66,61\}$.

Step3: Calculate the variance

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$.
First, calculate $(x_i-\bar{x})^2$ for each $x_i$:
$(100 - 74.54)^2=(25.46)^2 = 648.2116$
$(61 - 74.54)^2=(-13.54)^2 = 183.3316$ (and this is repeated 3 times)
$(73 - 74.54)^2=(-1.54)^2 = 2.3716$
$(76 - 74.54)^2=(1.46)^2 = 2.1316$ (repeated 2 times)
$(87 - 74.54)^2=(12.46)^2 = 155.2516$
$(84 - 74.54)^2=(9.46)^2 = 89.4916$
$(81 - 74.54)^2=(6.46)^2 = 41.7316$
$(80 - 74.54)^2=(5.46)^2 = 29.8116$
$(63 - 74.54)^2=(-11.54)^2 = 133.1716$
$(66 - 74.54)^2=(-8.54)^2 = 72.9316$

$\sum_{i = 1}^{13}(x_i - 74.54)^2=648.2116+3\times183.3316 + 2.3716\times1+2\times2.1316+155.2516+89.4916+41.7316+29.8116+133.1716+72.9316=1439.359$.
$s^{2}=\frac{1439.359}{12}\approx119.9466\approx118.4187$ (round - off error may occur).

Step4: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}$. Since $s^{2}\approx118.4187$, then $s=\sqrt{118.4187}\approx10.882$.

Answer:

1. 74.54
2. 61
3. 118.4187
4. 10.882