Adding data values

3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).

1. Find the mean and standard deviation of the 36 values.
2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.

2 Esme noted the test marks, \(x\), of 16 people in a class. She found that \(\Sigma x = 824\) and that the standard deviation of \(x\) was 6.5.

1. Calculate \(\Sigma ( x - 50 )\) and \(\Sigma ( x - 50 ) ^ { 2 }\).
2. One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.

6. Yujie is investigating the weights of 10 young rabbits. She records the weight, \(x\) grams, of each rabbit and the results are summarised below. $$\sum x = 8360 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 63840$$

1. Calculate the mean and the standard deviation of the weights of these rabbits. Given that the median weight of these rabbits is 815 grams,
2. describe, giving a reason, the skewness of these data. Two more rabbits weighing 776 grams and 896 grams are added to make a group of 12 rabbits.
3. State, giving a reason, how the inclusion of these two rabbits would affect the mean.
4. By considering the change in \(\sum ( x - \bar { x } ) ^ { 2 }\), state what effect the inclusion of these two rabbits would have on the standard deviation.

   END

1. Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
   0.0
   0.5
   1.8
   2.8
   2.3
   5.6
   9.4

Jenny then records the amount of rainfall, \(x \mathrm {~mm}\), at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$

1. Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
2. State, giving your reason, the effect this will have on the mean.

1. Each of the 25 students on a computer course recorded the number of minutes \(x\), to the nearest minute, spent surfing the internet during a given day. The results are summarised below.

$$\Sigma x = 1075 , \Sigma x ^ { 2 } = 44625 .$$

1. Find \(\mu\) and \(\sigma\) for these data. Two other students surfed the internet on the same day for 35 and 51 minutes respectively.
2. Without further calculation, explain the effect on the mean of including these two students.
   (2)

1. An adult evening class has 14 students. The ages of these students have a mean of 31.2 years and a standard deviation of 7.4 years.

A new student who is exactly 42 years old joins the class. Calculate the mean and standard deviation of the 15 students now in the group.

There are 10 numbers in a list. The first 9 numbers have mean 6 and variance 2. The 10th number is 3. Find the mean and variance of all 10 numbers. [6]

Eleven students in a class sit a Mathematics exam and their average score is 67% with a standard deviation of 12%. One student from the class is absent and sits the paper later, achieving a score of 85%.

1. Find the mean score for the whole class and the standard deviation for the whole class. [5]
2. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier. [2]

The marks of 24 students in a test had mean \(m\) and standard deviation \(\sqrt{6}\). Two new students took the same test. Their marks were \(m - 4\) and \(m + 4\). Show that the standard deviation of the marks of all 26 students is 2.60, correct to 3 significant figures. [3]