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1. The marks scored by some students are summarized in the following table:
Mark, $x$Frequency
$0 < x \leqslant 5$$5$
$5 < x \leqslant 10$$7$
$10 < x \leqslant 15$$8$
$15 < x \leqslant 20$$5$
  1. Find the modal class.
  2. Find an estimate for the mean.
  3. Explain why this is only an estimate.
2. A group of students took a test. The mean mark for $12$ boys was $18$, and the mean mark for $8$ girls was $16.5$. What is the mean mark for all $20$ students?
3. The salary of six teachers are (in thousands of pounds): $$24 \quad 29 \quad 33 \quad 35 \quad 62$$
  1. Find the mean and median salary of the teachers.
  2. The school wants to give the teachers an idea of the average amount they earn. Which of these two averages is more useful?
4. The numbers $a$, $b$ and $c$ have a median of $4$, a range of $7$, and a mean of $5.2$. Find the median, range and mean of $2a+1$, $2b+1$ and $2c+1$.
5. Given $$\sum x = 50 \quad \sum x^2 = 410 \quad n = 10$$ find the mean and the standard deviation.
6. The manager of a clothes shop recorded the size of each dress sold one morning to help decide what size dresses to order in the future: $$2\ 4\ 4\ 8\ 8\ 8\ 10\ 10\ 12\ 12\ 16\ 18$$
  1. Find the mean dress size sold and the standard deviation.
  2. Dress sizes are only in even numbers. Explain why the mean dress size is not very useful, and suggest a more useful average for the manager to use.
7. 80 students were asked to choose a number, $x$, between $0$ and $100$.
$x$Frequency
$0 < x \leqslant 25$$23$
$25 < x \leqslant 50$$18$
$50 < x \leqslant 75$$14$
$75 < x \leqslant 100$$25$

Use interpolation to find an estimate for the interquartile range of the numbers chosen.
8. The amount of time 200 students take to get to school in the mornings, $m$ minutes, were collected:
$m$Frequency
$0 < m \leqslant 10$$34$
$10 < m \leqslant 20$$53$
$20 < m \leqslant 30$$76$
$30 < m \leqslant 40$$37$

Calculate the $10$% to $90$% interpercentile range.
9. The heights of 10 girls, in cm, have the following summary statistics: $$\sum x = 1570 \quad \sum x^2 = 246900$$ The heights of 8 boys have the following summary statistics: $$\sum x = 1335 \quad \sum x^2 = 223275$$ Find the mean and standard deviation of all 18 children.
10. The number of detentions, $d$, 100 students received last term were recorded:
$d$Frequency
073
116
27
33
41

Calculate the mean and standard deviation.
11. The amount of pocket money, $p$ pounds, 80 students receive per week was recorded:
$p$Frequency
$0 < p \leqslant 5$$28$
$5 < p \leqslant 10$$24$
$10 < p \leqslant 15$$17$
$15 < p \leqslant 20$$11$

Estimate the number of students who earn more than 1 standard deviation above the mean amount.
12. The heights of a group of students, $x$ cm, were coded using the formula $$y = 5(x-165)$$ The summary statistics for $y$ are as follows: $$n = 10 \quad \sum y = 125 \quad \sum y^2 = 11875$$
  1. Find the mean and standard deviation of $y$.
  2. Hence find the mean and standard deviation of $x$.
13. The lengths of time each student in a class of $25$ spent chatting in a lesson were recorded.
The lower quartile of times was found to be $5$ minutes, and the upper quartile was found to be $12$ minutes.
An outlier is defined as any data point which lies more than $1$ interquartile range above the upper quartile, or below the lower quartile.
What are the largest and smallest amounts of time a student can chat in a lesson without being an outlier?
14. The number of after school clubs, $x$, at 100 schools are recorded.
$x$Frequency
$0 < x \leqslant 3$$15$
$3 < x \leqslant 6$$48$
$6 < x \leqslant 9$$37$

The data is coded using $$y = \frac{x - 3}{10}$$ Find an estimate for the mean and standard deviation of $y$.
15. The scores of 10 people who sat a maths test, $x$, are summarized: $$\sum x = 608 \quad \sum x^2 = 40034$$ An outlier is defined as any data point which lies outside of $2$ standard deviations of the mean.
  1. Find the smallest and highest test score that would not be considered an outlier.
  2. One of the 10 people who sat the test was the teacher, who scored $100$. Clean the data and find the mean and standard deviation of the 9 students who sat the test.
16. A school experimented with allowing half a class of $20$ students to use their phones in lessons. The scores of the students who used phones, $x$, and who did not use their phones, $y$, in an end of year test are summarized: $$\sum x = 557 \quad \sum x^2 = 39125$$ $$\sum y = 602 \quad \sum y^2 = 37934$$ By finding the mean and standard deviation, compare the effectiveness of the use of phones in lessons on test results.
17. The amount of time, $h$ hours, students spend on social media on a school night were recorded.
$h$Frequency
$0 < h \leqslant 1$$12$
$1 < h \leqslant 2$$31$
$2 < h \leqslant 3$$18$
$3 < h \leqslant 4$$14$

An outlier is defined as any data point which lies outside of $1$ standard deviation of the mean.
Estimate the number of outliers in this data.
18. A student calculated some statistics for a recent test their class of $20$ students did, and found that the mean mark was $62$, and the standard deviation was $28$. One student found that a page of their test was unmarked, and their score was increased from $46$ to $64$.
Find the new mean and standard deviation of the marks.