Statistics - Measures of dispersion & Central Tendency Flashcards
How do you create a frequency table using a calculator?
Menu setup > Statistics > 1: 1-Variable
Once values are imputed
OPTN > 3: 1-Variable Calc
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.
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.
How is the mean represented in statistics?
x̄ (x bar)
How is ‘sum of’ represented in statistics?
Σ (uppercase sigma)
What are measures of location? Give examples.
Single values which describe a POSITION in a data set.
Maximum
Minimum
Quartiles
Percentiles
Deciles
What are measures of central tendency? Give examples.
Values which describe the centre of the data, an ‘average’.
Mean
Median (the ones represented in a bell-curve)
Mode
What are measures of spread? Give examples.
How the data is spread out.
Standard deviation
Range
Interquartile range
What is the formula for working out the mean of ungrouped data?
x̄ = Σx / n
or
x̄ = (Σfx) / Σf
How would you find an estimate of the mean using grouped data?
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.
What is the formula for linear interpolation to find the median?
median (Q2) - lowerbound position of median - CF of group before
━━━━━━━━━━━━━ = ━━━━━━━━━━━━━━━━━━━━
group width CF of median group - CF of group before
What would be the symbol to represent the 60th percentile?
P₆₀
How would you work out the median of grouped data? n = 200
1 > n x 1/2 = 100th position
2 > Find the class with the 100th position included in its CF.
3 > Use linear interpolation formula
How would you work out the 60th percentile of grouped data? n = 200
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
How would you work out the upper or lower quartile of grouped data? n = 200
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
What is the symbol for variance?
σ²
sigma squared
What is the formula to work out the variance?
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
What is the symbol for standard deviation?
σ
lowercase sigma
What is the formula for working out the standard deviation?
Square root of σ² (variance)
What are the rules of coding?
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
What will the effect of this code…
y = x + 10
have on the mean ( x̄ ) and the standard deviation ( σ ) ?
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)
What will the effect of this code…
y = 3x
have on the mean ( x̄ ) and the standard deviation ( σ ) ?
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)
What will the effect of this code…
y = 2x - 5
have on the mean ( x̄ ) and the standard deviation ( σ ) ?
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.
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
New mean ( ȳ ) = 40
New standard deviation ( σᵧ ) = 5
