Question

find the mean, median, mode, and standard deviation of the following data set: {1,2,2,2,3,3,4,4,5} (up to 2 decimal places) mean: 2.89 median: 3 mode: 2 (sample) standard deviation:

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{1 + 2+2+2+3+3+4+4+5}{9}=\frac{26}{9}\approx2.89$

Step2: Find the median

The data set has 9 (odd - numbered) elements. Arranged in ascending order, the middle - numbered element is the 5th one. So the median is 3.

Step3: Determine the mode

The mode is the number that appears most frequently. Here, 2 appears 3 times, more frequently than other numbers, so the mode is 2.

Step4: Calculate the sample standard deviation

First, find the differences from the mean:
$d_1=1 - 2.89=-1.89$, $d_2 = 2-2.89=-0.89$, $d_3 = 2-2.89=-0.89$, $d_4 = 2-2.89=-0.89$, $d_5 = 3-2.89 = 0.11$, $d_6 = 3-2.89 = 0.11$, $d_7 = 4-2.89 = 1.11$, $d_8 = 4-2.89 = 1.11$, $d_9 = 5-2.89 = 2.11$
Then square the differences:
$d_1^2=(-1.89)^2 = 3.5721$, $d_2^2=(-0.89)^2 = 0.7921$, $d_3^2=(-0.89)^2 = 0.7921$, $d_4^2=(-0.89)^2 = 0.7921$, $d_5^2=(0.11)^2 = 0.0121$, $d_6^2=(0.11)^2 = 0.0121$, $d_7^2=(1.11)^2 = 1.2321$, $d_8^2=(1.11)^2 = 1.2321$, $d_9^2=(2.11)^2 = 4.4521$
The sum of squared differences $\sum_{i = 1}^{9}d_i^2=3.5721+3\times0.7921 + 2\times0.0121+2\times1.2321+4.4521=3.5721 + 2.3763+0.0242+2.4642+4.4521 = 12.889$
The sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=\sqrt{\frac{12.889}{9 - 1}}=\sqrt{\frac{12.889}{8}}\approx1.27$

Answer:

Mean: 2.89
Median: 3
Mode: 2
(Sample) Standard Deviation: 1.27