Question

1. calculate the mean of the data 9, 10, 12, 13, 13, 13, 15, 15, 16, 16, 18, 22, 23, 24, 24, 25

Explanation:

Step1: Sum all the data points

First, we add up all the numbers:
\(9 + 10 + 12 + 13 + 13 + 13 + 15 + 15 + 16 + 16 + 18 + 22 + 23 + 24 + 24 + 25\)
Let's calculate step by step:
\(9+10 = 19\)
\(19+12 = 31\)
\(31 + 13\times3=31 + 39 = 70\)
\(70+15\times2 = 70 + 30 = 100\)
\(100+16\times2 = 100 + 32 = 132\)
\(132+18 = 150\)
\(150+22 = 172\)
\(172+23 = 195\)
\(195+24\times2 = 195 + 48 = 243\)
\(243+25 = 248\)? Wait, no, let's recalculate correctly:

Wait, let's list all numbers: 9, 10, 12, 13, 13, 13, 15, 15, 16, 16, 18, 22, 23, 24, 24, 25.

Count the number of terms: Let's count: 9 (1), 10 (2), 12 (3), 13 (4,5,6), 15 (7,8), 16 (9,10), 18 (11), 22 (12), 23 (13), 24 (14,15), 25 (16). So total \(n = 16\) terms.

Now sum:

9 + 10 = 19

19 + 12 = 31

31 + 13 + 13 + 13 = 31 + 39 = 70

70 + 15 + 15 = 70 + 30 = 100

100 + 16 + 16 = 100 + 32 = 132

132 + 18 = 150

150 + 22 = 172

172 + 23 = 195

195 + 24 + 24 = 195 + 48 = 243

243 + 25 = 268? Wait, I must have miscalculated earlier. Let's use a better way:

List all numbers:

9, 10, 12, 13, 13, 13, 15, 15, 16, 16, 18, 22, 23, 24, 24, 25

Let's group them:

Single terms: 9, 10, 12, 18, 22, 23, 25

Triple terms: 13 (three times)

Double terms: 15 (two), 16 (two), 24 (two)

Sum of single terms: \(9 + 10 + 12 + 18 + 22 + 23 + 25\)
\(9+10=19\); \(19+12=31\); \(31+18=49\); \(49+22=71\); \(71+23=94\); \(94+25=119\)

Sum of triple 13s: \(13\times3 = 39\)

Sum of double 15s: \(15\times2 = 30\)

Sum of double 16s: \(16\times2 = 32\)

Sum of double 24s: \(24\times2 = 48\)

Total sum: \(119 + 39 + 30 + 32 + 48\)
\(119+39=158\); \(158+30=188\); \(188+32=220\); \(220+48=268\)

Step2: Calculate the mean

The mean \(\bar{x}\) is given by the formula \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n\) is the number of data points and \(\sum_{i = 1}^{n}x_{i}\) is the sum of the data points.

We found that the sum \(\sum x_{i}=268\) and \(n = 16\).

So, \(\bar{x}=\frac{268}{16}\)
Simplify \(\frac{268}{16}=\frac{67}{4}=16.75\)

Answer:

The mean of the data is \(16.75\)