Statistics

Have a go first. Then open the working one line at a time.

1. Add

   35 + 40 + 25 + 45 + 30 + 40 + 150 = 365

   Add every value.

2. Mean

   365 ÷ 7 = 52.14

   Divide by the number of values. Two decimal places.

3. Order, then median

   25, 30, 35, 40, 40, 45, 150: median = 40

   Put the values in order. With 7 values the median is the 4th.

4. Compare

   mean 52.14 and median 40

   The 150 pulls the mean above all but one of the values. Six of the seven students spent 45 minutes or less, so the median describes a typical student better.

Have a go first. Then open the working one line at a time.

1. Order the list

   14, 16, 17, 19, 20, 21, 22, 24, 25, 28, 31

   Quartiles need an ordered list.

2. Quartiles

   From the calculator: min 14, Q1 17, median 21, Q3 25, max 31

   n = 11 is odd. The calculator leaves the middle value out, then takes the median of each half.

3. Five-number summary

   14, 17, 21, 25, 31

   Write min, Q1, median, Q3, max in that order.

4. IQR

   IQR = Q3 − Q1 = 25 − 17 = 8

   The spread of the middle half of the data.

Have a go first. Then open the working one line at a time.

1. Route A

   IQR = 10 − 4 = 6

   Q3 − Q1.

2. Route B

   IQR = 8 − 5 = 3

   Same rule.

3. Compare

   Both medians are 7 min. IQR: A 6, B 3.

   The typical delay is the same. Route A's middle half is spread over 6 minutes, route B's over 3, so route B is more consistent.

Have a go first. Then open the working one line at a time.

1. IQR

   IQR = 59 − 41 = 18

   Fences start from the IQR.

2. 1.5 × IQR

   1.5 × 18 = 27

   This is the distance beyond each quartile.

3. Fences

   Lower: 41 − 27 = 14. Upper: 59 + 27 = 86

   A value is an outlier only if it is beyond a fence.

4. Test the values

   10 < 14: outlier. 80 is between 14 and 86: not. 90 > 86: outlier.

   Compare each value with both fences.

Have a go first. Then open the working one line at a time.

1. Positions

   \(n=40\): median between the 20th and 21st values, \(Q_1\) between the 10th and 11th, \(Q_3\) between the 30th and 31st

   The table stands for a list of 40 values in order.

2. Cumulative frequency

   Running totals: 6, 17, 26, 33, 37, 39, 39, 40

   The first value whose total reaches a position is the value at that position.

3. Quartiles

   From the calculator: \(Q_1=1\), median \(=2\), \(Q_3=3\)

   One-variable statistics with the values in one list and the frequencies in the other gives the same.

4. Interquartile range

   \(IQR=3-1=2\)

   Q3 minus Q1.

5. Outlier test

   \(3+1.5\times 2=6\), and \(7>6\)

   7 is above the upper fence, so a household with 7 pets is an outlier.

Have a go first. Then open the working one line at a time.

1. Quartiles

   From the calculator: min 5, Q1 9, median 12.5, Q3 16, max 41

   n = 12 is even: each half has 6 values.

2. Fences

   IQR = 7; 1.5 × 7 = 10.5. Lower: −1.5. Upper: 26.5.

   Q1 − 1.5 IQR and Q3 + 1.5 IQR.

3. Outliers

   41 > 26.5: outlier. No value is below −1.5.

   Check the smallest and the largest values.

4. Whiskers

   Lower whisker to 5. Upper whisker to 18, the largest value inside the fence.

   A whisker stops at the last value that is not an outlier.

5. Draw it

   Box from 9 to 16, median line at 12.5, cross at 41.

   Scale the axis to include the outlier.

Have a go first. Then open the working one line at a time.

1. Mean

   mean = 40 ÷ 5 = 8

   Start with the mean.

2. Deviations

   Subtract the mean from each value, then square.

   Squaring makes every term positive.

3. Variance

   sum = 34; variance = 34 ÷ 5 = 6.8

   Divide by n, the number of values.

4. Square root

   σ = √6.8 = 2.61

   The standard deviation is the square root of the variance.

Have a go first. Then open the working one line at a time.

1. Enter the data

   Type the 10 times in cells A1 to A10.

   One value per cell.

2. Mean

   =AVERAGE(A1:A10) gives 28.96

   Round to 2 decimal places.

3. Standard deviation

   =STDEV.P(A1:A10) gives 2.76

   STDEV.P is the population standard deviation. STDEV.S is the sample one: not this.