Nota Mathematics Tingkatan 4 Bab 8

Measures of dispersion show how the values of a set of data are scattered. Dispersion is small if the data set has a small range, and vice versa. (p.212)
Measures of dispersion: range, interquartile range, variance and standard deviation. (p.212)
To represent data ethically: use a uniform scale that starts from 0, and display accurate data. (p.212)
Large difference between largest and smallest values means data is widely dispersed; a small difference means narrow dispersion. (p.213)
Stem-and-leaf plot: shows the distribution of data; tens digit as the stem and units digit as the leaf, with leaf digits arranged from smallest to largest; include a key. (p.214)
Dot plot: points plotted on a uniform scale, each point represents a value; suitable for small data sets. (p.215)
Compare two stem-and-leaf plots (back to back) by their patterns; more data at higher stems means better achievement. (p.216)
Compare two dot plots by shape, dispersion of points, frequencies, and whether values are concentrated on the left or right. (p.217)

Range = Largest value − Smallest value. (p.219)
From a frequency table: range = largest score − smallest score. (p.219)
Interquartile range = Q₃ − Q₁. (p.220)
Steps: arrange data in ascending order; find the median; split into lower and upper halves; Q₁ = middle of the lower half; Q₃ = middle of the upper half. (p.220)
If the number of values is even, the median/Q₁/Q₃ is the average of the two middle values. Second quartile Q₂ = median. (p.220)
Q₁ is the lower quartile; Q₃ is the upper quartile. (p.220)
Frequency table: use cumulative frequency; Q₁ = the (¼ × N)th value; Q₃ = the (¾ × N)th value. (p.221)

Variance is the average of the squares of the differences between each value and the mean. Standard deviation is the square root of the variance, in the same unit as the data. (p.221)
Mean x̄ = Σx / N. (p.221)
Variance σ² = Σ(x − x̄)² / N = Σx² / N − x̄². (p.221)
Standard deviation σ = √[Σ(x − x̄)² / N] = √[Σx² / N − x̄²]. (p.221)
Frequency table: x̄ = Σfx / Σf; σ² = Σf(x − x̄)² / Σf = Σfx² / Σf − x̄²; σ = √σ². (p.222)
Calculator: MODE → SD, enter each value with M+ (repeat for frequency), then SHIFT 2 for x̄ or xσn; square the standard deviation to get the variance. (p.223)

Range is the easiest to calculate but does not give a good overview of how data is distributed. (p.224)
If there is an outlier or extreme value, the interquartile range is the more appropriate measure of dispersion. (p.224)
Standard deviation is usually used to compare two sets of data: a low standard deviation means data is close to the mean; a high one means data is far from the mean. (p.224)
The set with the smaller standard deviation is more consistent. (p.225)
Comparing data sets: measures of central tendency alone may not be enough; also compare measures of dispersion. (p.234)

A box plot shows the distribution of data using five values: minimum value, first quartile, median, third quartile and maximum value. (p.226)
A box plot shows whether the data is symmetric about the median and is often used to analyse a large number of data. (p.226)
Construct on a number line: mark the five values, draw the box from Q₁ to Q₃ with the median inside, and whiskers to the minimum and maximum. It can also be drawn vertically. (p.226)
From a box plot: interquartile range = Q₃ − Q₁; range = maximum − minimum. (p.227)

Data changes include: adding/subtracting a constant, multiplying/dividing by a constant, outliers, and adding or removing values. (p.228)
Add or subtract a constant to every value: range, interquartile range, variance and standard deviation do not change. (p.228)
Multiply every value by k: new range = k × range; new IQR = k × IQR; new standard deviation = k × σ; new variance = k² × variance. (p.229)
Divide every value by k: new range, IQR and standard deviation are the originals divided by k; new variance = original variance / k². (p.229)
Adding or removing an outlier: range changes dramatically; interquartile range is less affected; variance and standard deviation increase significantly when an outlier is added. (p.229)
On a dot plot, adding a constant shifts the whole plot with the same dispersion. (p.230)
On a dot plot, multiplying by 2 makes the dispersion twice as wide (new range = k × original range). (p.231)
Adding an outlier makes the dispersion wider; the measures of dispersion become larger. (p.232)
Adding a value far from the mean increases the standard deviation; adding a value close to the mean decreases it. Removing a value far from the mean decreases it. (p.233)

Use mean and standard deviation to decide which data set is more efficient or consistent (smaller standard deviation). (p.235)
Combining groups: recover Σx = N × mean and Σx² = N(σ² + x̄²) for each group, add them, then recompute the mean and standard deviation. (p.236)

Chapter 8: Measures of Dispersion for Ungrouped Data