Statistics - Measures of dispersion & Central Tendency

Question 1

How do you create a frequency table using a calculator?

Answer 1

Menu setup > Statistics > 1: 1-Variable

Once values are imputed

OPTN > 3: 1-Variable Calc

Question 2

The mean maths score of 20 pupils in class A is 62.
The mean maths score of 30 pupils in class B is 75.

a) What is the overall mean of all the pupils’ marks.

b) The teacher realises they mis-marked one student’s paper; he should have received 100 instead of 95. Explain the effect on the mean and median.

Answer 2

a) THINK TOTALS !!!

20 x 62 =1240
30 x 75 = 2250

2250 + 1240 = 3490

3490 / 50 = 69.8

b) The score will increase the mean but the median remains the same.

Question 3

How is the mean represented in statistics?

Answer 3

x̄ (x bar)

Question 4

How is ‘sum of’ represented in statistics?

Answer 4

Σ (uppercase sigma)

Question 5

What are measures of location? Give examples.

Answer 5

Single values which describe a POSITION in a data set.

Maximum
Minimum
Quartiles
Percentiles
Deciles

Question 6

What are measures of central tendency? Give examples.

Answer 6

Values which describe the centre of the data, an ‘average’.

Mean
Median (the ones represented in a bell-curve)
Mode

Question 7

What are measures of spread? Give examples.

Answer 7

How the data is spread out.

Standard deviation
Range
Interquartile range

Question 8

What is the formula for working out the mean of ungrouped data?

Answer 8

x̄ = Σx / n

or

x̄ = (Σfx) / Σf

Question 9

How would you find an estimate of the mean using grouped data?

Answer 9

1# Ensure there are no gaps between each inequality…
E.G
0 ≤ h < 0.5 <= THIS 0 ≤ h < 0.5 (There are values between 0.5 and 0.6
0.5 ≤ h < 1.2 NOT THIS => 0.6 ≤ h < 1.2 that are not included)

2# If there are gaps, close them. 0 ≤ h < 0.55
0.55 ≤ h < 1.2

3# Find the midpoint of each inequality, then proceed with…
x̄ = (Σfx) / Σf

or input values in statistics mode to produce a frequency table.

Question 10

What is the formula for linear interpolation to find the median?

Answer 10

median (Q2) - lowerbound position of median - CF of group before
━━━━━━━━━━━━━ = ━━━━━━━━━━━━━━━━━━━━
group width CF of median group - CF of group before

Question 11

What would be the symbol to represent the 60th percentile?

Answer 11

P₆₀

Question 12

How would you work out the median of grouped data? n = 200

Answer 12

1 > n x 1/2 = 100th position

2 > Find the class with the 100th position included in its CF.

3 > Use linear interpolation formula

Question 13

How would you work out the 60th percentile of grouped data? n = 200

Answer 13

1 > n x 0.6 = 120th position (60% of 200)

2 > Find the class with the 120th position included in its CF.

3 > Use linear interpolation formula

Question 14

How would you work out the upper or lower quartile of grouped data? n = 200

Answer 14

1 >
LQ: 200 x 1/4 = 50th position
UQ: 200 x 3/4 = 150th position

2 > Find the classes with the 50th and 150th position included in its CF.

3 > Use linear interpolation formula

Question 15

What is the symbol for variance?

Answer 15

σ²

sigma squared

Question 16

What is the formula to work out the variance?

Answer 16

        Σfx²         ( Σfx )² σ² =    ━━━  -   ━━━          msmsm = (the mean of squares minus the square of mean)
          Σf           (  Σf  )²  

OR

       Σx² σ² =   ━━  - ( x̄ )²         ( the second part of the 1st equation is the same as x̄ )
        n

Question 17

What is the symbol for standard deviation?

Answer 17

σ

lowercase sigma

Question 18

What is the formula for working out the standard deviation?

Answer 18

Square root of σ² (variance)

Question 19

What are the rules of coding?

Answer 19

When every x value are changed…

1 = adding / subtracting has an affect on the mean but NO effect on SD

2 = multiplication or division affects both the mean and SD

Question 20

What will the effect of this code…

y = x + 10

have on the mean ( x̄ ) and the standard deviation ( σ ) ?

Answer 20

Suppose our variable (e.g. heights in cm) was x.
Then y would represent the heights with 10cm added on to each value.

x̄ will similarly increase by 10.

σ will remain the SAME.

(No effect on SD from adding/subtracting)

Question 21

What will the effect of this code…

y = 3x

have on the mean ( x̄ ) and the standard deviation ( σ ) ?

Answer 21

Suppose our variable (e.g. heights in cm) was x.
Then y would represent the heights with 10cm added on to each value.

x̄ will get 3 times bigger.

σ will also get 3 times bigger.

(SD only affected by multiplication/division)

Question 22

What will the effect of this code…

y = 2x - 5

have on the mean ( x̄ ) and the standard deviation ( σ ) ?

Answer 22

Suppose our variable (e.g. heights in cm) was x.
Then y would represent the heights with 10cm added on to each value.

x̄ will be multiplied by 2 and subtracted by 5.

σ will get 2 times larger.

Question 23

If the old mean ( x̄ ) = 300,
and the old SD ( σ ) = 25…

What would the new mean ( ȳ ) and new SD ( σᵧ ) be if the coding was…

      x - 100 y =   ━━━━
          5

Answer 23

New mean ( ȳ ) = 40

New standard deviation ( σᵧ ) = 5