Describing Location in a Distribution - Study Notes

Definition: The p-th percentile of a distribution is the value with p percent of the observations less than it.
- Expressed mathematically: the p-th percentile is the value xp such that the proportion of observations less than xp is p/100.
Example (Percentile interpretation): Jenny earned a score of 86 on her test.
- If 21 of the 25 observations are below hers, then Jenny is at the 84th percentile because
  $\frac{21}{25} = 0.84 = 84\%.$
- This means her score is higher than 21 out of 25 observations.
Takeaways:
- Percentiles describe location within a distribution.
- They are useful for comparing an individual value to the overall distribution.

Purpose: A cumulative relative frequency graph displays the cumulative relative frequency of each class in a frequency distribution.
Example data: Age at inauguration for the 44 Presidents
- Classes: 40-44, 45-49, 50-54, 55-59, 60-64, 65-69
- Frequencies: 2, 7, 13, 12, 7, 3 respectively
- Relative frequencies (RF):
- 40-44: $\frac{2}{44} = 0.045$ (4.5%)
- 45-49: $\frac{7}{44} = 0.159$ (15.9%)
- 50-54: $\frac{13}{44} = 0.295$ (29.5%)
- 55-59: $\frac{12}{44} = 0.340$ (34.0%)
- 60-64: $\frac{7}{44} = 0.159$ (15.9%)
- 65-69: $\frac{3}{44} = 0.068$ (6.8%)
- Cumulative frequencies (CF):
- 40-44: 2; Cumulative relative frequency (CRF): 0.045
- 45-49: 9; CRF: 0.205
- 50-54: 22; CRF: 0.500
- 55-59: 34; CRF: 0.773
- 60-64: 41; CRF: 0.932
- 65-69: 44; CRF: 1.000
- Notes:
- The CRF at each class gives the proportion of observations up to the upper boundary of that class.
- Useful for reading approximate percentiles directly from the graph.

Definition: A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
Formula: if an observation is x from a distribution with mean $\mu$ and standard deviation $\sigma$ , then $z = \frac{x - \mu}{\sigma}.$
- The z-score is often denoted as z and called a standardized score.
Example: Jenny scored 86; mean = 80; standard deviation = 6.07
- Calculation:
  $z = \frac{86 - 80}{6.07} \approx 0.99.$
- Interpretation: Jenny’s score is about 0.99 standard deviations above the mean.

Purpose: Transformations convert data to a different scale and can affect the distribution’s shape, center, and spread.
General ideas:
- Adding or subtracting a constant a to every observation:
- Increases or decreases measures of center and location by a: e.g., mean, median, quartiles, percentiles shift by a.
- Does not change the shape of the distribution or the measures of spread (range, IQR, standard deviation).
- Multiplying or dividing every observation by a constant b:
- Multiplies (or divides) measures of center and location by b.
- Multiplies (or divides) measures of spread (range, IQR, standard deviation) by |b|.
- Does not change the shape of the distribution.
Mathematical summary:
- If you add a: $\text{mean}{new} = \text{mean}{old} + a,\quad \text{median}{new} = \text{median}{old} + a,\quad \text{Q1}{new} = \text{Q1}{old} + a,\quad \text{Q3}{new} = \text{Q3}{old} + a.$
- If you multiply by b: $\text{mean}{new} = b\,\text{mean}{old},\quad \text{sd}{new} = |b|\,\text{sd}{old},\quad \text{IQR}{new} = |b|\,\text{IQR}{old},\quad \text{Range}{new} = |b|\,\text{Range}{old}.$
Example: Defining a new variable error = guess − 13
- This is a shift of all data by −13 units.
- Shape of the distribution remains the same; center/location decreases by 13; spread remains the same.
Example: Multiplying by a constant b
- If you multiply each observation by b, all location measures are scaled by b and all spread measures by |b|, with the distribution’s shape preserved.

Context: Australian students’ data converted from meters to feet (1 meter ≈ 3.28 feet) to report back in familiar units.
General rule for linear conversion:
- If y = c x with c = 3.28, then
- Mean in feet = 3.28 × mean in meters
- Standard deviation in feet = 3.28 × sd in meters
- Min, Q1, Median, Q3, Max, IQR, Range all scale by 3.28 as well.
Given data (Error in meters vs. feet):
- Error (m): n = 44; mean = 3.02; sd = 7.14; Min = -5; Q1 = -2; Median = 2; Q3 = 4; Max = 27; IQR = 6; Range = 32
- Error (ft): n = 44; mean = 9.91; sd = 23.43; Min = -16.4; Q1 = -6.56; Median = 6.56; Q3 = 13.12; Max = 88.56; IQR = 19.68; Range = 104.96
Verification via conversion:
- 3.28 × mean(m) = 3.28 × 3.02 ≈ 9.91
- 3.28 × sd(m) = 3.28 × 7.14 ≈ 23.43
- 3.28 × Min(m) = 3.28 × (-5) = -16.4
- 3.28 × Q1(m) = 3.28 × (-2) = -6.56
- 3.28 × Median(m) = 3.28 × 2 = 6.56
- 3.28 × Q3(m) = 3.28 × 4 = 13.12
- 3.28 × Max(m) = 3.28 × 27 = 88.56
- 3.28 × IQR(m) = 3.28 × 6 = 19.68
- 3.28 × Range(m) = 3.28 × 32 = 104.96
Takeaway: Unit conversions are linear transformations that scale both the center and spread by the same factor, preserving the distribution’s shape.

You should be able to:
- FIND and INTERPRET the percentile of an individual value within a distribution.
- ESTIMATE percentiles and individual values using a cumulative relative frequency graph.
- FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution.
- DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution.
Key formulas to remember:
- Percentiles: definition described above.
- Cumulative relative frequency: cumulative percentages up to each class.
- z-score: $z = \frac{x - \mu}{\sigma}$
- Transformations:
- Addition/subtraction: center changes by a; spread unchanged.
- Multiplication/division: center and spread scale by the factor; shape unchanged.
Real-world relevance:
- Percentiles help compare individuals to a group.
- z-scores standardize data for cross-distribution comparisons.
- Transformations are essential when changing units or shifting datasets for analysis.