Measures of location and dispersion flashcards

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Measure of location
A single value representing the centre or a typical value of the data: mean, median or mode.
Median
The middle value of the ordered data (Q₂); half the values are below it and half above.
Modal class
The class with the highest frequency density (the highest frequency if the classes have equal widths).
Interquartile range
Q₃ − Q₁: the spread of the middle 50% of the data. Not affected by extreme values.
Standard deviation
The square root of the variance: a measure of the typical distance of the values from the mean, in the units of the data.
Class boundaries
The true limits of a class. Data to the nearest whole number in the class 15–19 have boundaries 14.5 and 19.5 (width 5).
10th to 90th interpercentile range
P₉₀ − P₁₀: the spread of the middle 80% of the data.
Mean of a frequency table
x̄ = Σfx ÷ Σf (use class midpoints for grouped data). (No — learn it)
Variance of data from Σx and Σx²
σ² = Σx²/n − x̄² (mean of the squares minus the square of the mean). (No — learn it)
Variance of a frequency table
σ² = Σfx²/Σf − x̄². (No — learn it)
Sxx
Sxx = Σx² − (Σx)²/n, and variance = Sxx/n. (Yes)
Coding y = (x − a)/b: effect on mean and s.d.
ȳ = (x̄ − a)/b and σ_y = σ_x/b. Decode: x̄ = bȳ + a, σ_x = bσ_y. (No — learn it)
Interpolation for the median of grouped data
L + ((n/2 − C)/f_c) × w: L lower boundary, C cumulative frequency below L, f_c class frequency, w class width. (No — derive it)
Position of Q₁, Q₂, Q₃ in a list of n values
n/4, n/2, 3n/4: not a whole number → round up; whole number k → mean of the kth and (k + 1)th values. (No — learn it)
Two dot plots with the same mean, one tightly packed and one spread out
Same location, different dispersion: the spread-out set has the larger standard deviation and IQR.
Cumulative frequency polygon and interpolation
Plot cumulative frequency at upper class boundaries, join with straight lines; reading across from n/2 gives the interpolated median.
Two box plots on one scale with similar medians but different box lengths
Similar typical values; the longer box (larger IQR) shows less consistent data.
Coding y = (x − a)/b on two number lines
Subtracting a slides the data along (mean moves, spread unchanged); dividing by b shrinks every gap by a factor b.
Distribution with a long tail of high values
The mean is pulled above the median towards the tail.
When are the median and IQR better than the mean and s.d.?
When there are extreme values or the data are not symmetrical: they are not affected by how extreme the end values are.
Why are the mean and s.d. of grouped data only estimates?
The individual values are unknown; each is assumed to sit at its class midpoint.
Why does adding a constant not change the standard deviation?
Every value and the mean move by the same amount, so the distances from the mean are unchanged.
Effect of adding a value equal to the mean
The mean is unchanged; the s.d. decreases (Sxx is unchanged but n increases).
How do you combine two groups’ means and standard deviations?
Recover Σx = n x̄ and Σx² = n(σ² + x̄²) for each, add the sums and the n’s, then use the formulae.
Why can a variance never be negative?
It is the mean of squared distances from the mean. A negative answer means an arithmetic slip.
n = 25, Σx = 1150, Σx² = 54 610. Mean and s.d.?
x̄ = 46; σ² = 2184.4 − 2116 = 68.4; σ = 8.27.
Quartiles of 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 8, 9, 12, 15, 27?
n = 15: Q₁ = 4th = 2, median = 8th = 5, Q₃ = 12th = 9; IQR = 7.
n = 70; 22 values below 15; the class 15–20 has frequency 21. Median?
Position 35: 15 + (13/21) × 5 = 18.1.
y = (x − 200)/5, n = 20, Σy = 34, Σy² = 182. Mean and s.d. of x?
ȳ = 1.7, σ_y² = 6.21; x̄ = 5(1.7) + 200 = 208.5, σ_x = 5√6.21 = 12.5.
Group A: n = 30, mean 62, s.d. 5. Group B: n = 20, mean 70, s.d. 4. Combined mean and s.d.?
Σx = 3260, Σx² = 214 390; mean 65.2; σ² = 4287.8 − 65.2² = 36.76; σ = 6.06.
Temperatures: mean 18.4 °C, s.d. 3.5 °C. Mean and s.d. in °F (F = 1.8C + 32)?
Mean 1.8 × 18.4 + 32 = 65.12 °F; s.d. 1.8 × 3.5 = 6.3 °F.
x = 0,1,2,3,4,5 with f = 6,11,9,7,4,3. Mode, median, mean?
Mode 1; median = mean of the 20th and 21st values = 2; mean = 81/40 = 2.025.
Squaring the total instead of totalling the squares
(Σx)² ≠ Σx². Variance = Σx²/n − x̄², with Σx² the sum of the squared values.
Decoding the standard deviation
Only the scale factor applies: σ_x = bσ_y. Never add a back on; the variance uses b².
Interpolating with class limits instead of boundaries
For data to the nearest unit, use boundaries such as 54.5–59.5 (width 5), not 55–59 (width 4).
“Compare the two distributions”
Compare a location and a spread, each with numbers and an interpretation in context.
Rounding the mean before squaring it
Keep x̄ exact or at full calculator accuracy in σ² = Σx²/n − x̄²; round only the final answer (3 s.f.).
A value was recorded wrongly
Correct both sums: Σx − wrong + right and Σx² − wrong² + right², then recompute.

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