STATISTICS for bio

STATISTICS 1403

PROBABILITY AND STATISTICS FOR THE BIOSCIENCES

PROBLEM SET: MEAN, MEDIAN, AND MODE

Abstract. How are the mean, median, and mode related for right-skewed distributions?

1. Introduction

The lognormal, Weibull, and Rayleigh distributions are all right-skewed distributions of non-negative random variables (x ≥ 0). Their PDFs, CDFs, and means are given here:

distribution PDF CDF mean

lognormal f(x; µ,σ) = 1 σx √ 2π e−(ln x−µ)

2/2σ2 F(x) = Φ (

ln x−µ σ

) eµ+σ

2/2

Rayleigh f(x; σ) = x σ2 e−x

2/2σ2 F(x) = 1 − e−x 2/2σ2 σ

√ π 2

Weibull f(x; k,σ) = k σ

( x σ

)k−1 e−(x/σ)

k

F(x) = 1 − e−(x/σ) k

σ Γ ( 1 + 1

k

)

2. Medians and Modes

The median of a distribution is the x value where the CDF equals one-half

F(xm) = 0.5

while the mode is the most frequent value, where the PDF is maximized

d

dx f(x) = 0

(a) Calculate the median for each distribution, using a scale parameter σ = 1.

(b) Calculate the mode for each distribution, using a scale parameter σ = 1.

(c) For each distribution, place the mean1, median, and mode in increasing order. What do you notice?

For all calculations, assume the lognormal parameter µ = 1 and the Weibull paramater k = 2.

1For the Weibull mean, the gamma function, Γ() can be calculated using MATLAB.

1

2 MEAN, MEDIAN, AND MODE

3. Your Analysis

Your analysis must be legibly handwritten. Include a brief problem statement (in your own words, do NOT copy these instructions), and a brief narrative explaining your calculations and findings. BE SURE TO PUT YOUR NAME ON THE WRITE-UP. Submit your write-up via Blackboard as a scanned or saved .PDF or Word .DOC file.