Estimating with Uncertainty
M. Drew LaMar
September 15, 2021
“I believe that we do not know anything for certain, but everything probably.”
- Christiaan Huygens
Course Announcements
- Reading Assignment for Friday - W&S, Chapter 5 (
QUIZ)
Moving on...
Make sure you read the book for the following discussions
- How to compute a mean and standard deviation from a frequency table
Question: Why is this important to know?
- Rounding rules for displaying tables and statistics
- Effect of changing measurement scale
- Cumulative frequency distributions (we will cover this later as well)
My point here is that you are responsible for all book material, even if we don't cover it in lecture!
Describing data in R
| Measures | R commands |
|---|---|
| \( \overline{Y} \) | mean |
| \( s^2 \) | var |
| \( s \) | sd |
| \( IQR \) | IQR\( ^* \) |
| Multiple | summary |
\( ^* \) Note that IQR has different algorithms. To match the algorithm in W&S, you should use IQR(___, type=5). There are different algorithms as there are different ways to calculate quantiles. (for curious souls, see ?quantiles). For the HW, either version is acceptable. Default type in R is type=7.
Describing data in R
| Measures | R commands |
|---|---|
| \( \overline{Y} \) | mean |
| \( s^2 \) | var |
| \( s \) | sd |
| \( IQR \) | IQR |
| Multiple | summary |
summary(mydata)
breadth
Min. : 1.00
1st Qu.: 3.00
Median : 8.00
Mean :11.88
3rd Qu.:17.00
Max. :62.00
IQR would be \( 17-3 = 14 \).
Precision vs Accuracy
Language: Sampling Distributions
Definition: The
sampling distribution is the population distribution of all values for an estimate that we might obtain when we sample a population.
Definition: The
standard error of an estimate is the standard deviation of the estimate’s sampling distribution.
Definition: The
standard error of the mean is given by
\[ \sigma_{\overline{Y}} = \frac{\sigma}{\sqrt{n}} \] with theapproximate standard error of the mean given by \[ \mathrm{SE}_{\overline{Y}} = \frac{s}{\sqrt{n}} \]
"Chalk" talk - Sampling distributions and 95% confidence intervals
Language: Confidence Intervals
Definition: A
confidence interval is a range of values surrounding the sample estimate that is likely to contain the population parameter.
Definition: A
95% confidence interval provides a most-plausible range for a parameter. Values lying within the interval are most plausible, whereas those outside are less plausible, based on the data.
Error bars
How to do these in R?
Read and inspect the data.
locustData <- read.csv("../..//Datasets/chapter02/chap02f1_2locustSerotonin.csv")
head(locustData)
serotoninLevel treatmentTime
1 5.3 0
2 4.6 0
3 4.5 0
4 4.3 0
5 4.2 0
6 3.6 0
str(locustData)
'data.frame': 30 obs. of 2 variables:
$ serotoninLevel: num 5.3 4.6 4.5 4.3 4.2 3.6 3.7 3.3 12.1 18 ...
$ treatmentTime : int 0 0 0 0 0 0 0 0 0 0 ...
Error bars
First, calculate the statistics by group needed for the error bars: the mean and standard error. Here, tapply is used to obtain each quantity by treatment group.
meanSerotonin <- tapply(locustData$serotoninLevel,
locustData$treatmentTime,
mean)
sdSerotonin <- tapply(locustData$serotoninLevel,
locustData$treatmentTime,
sd)
nSerotonin <- tapply(locustData$serotoninLevel,
locustData$treatmentTime,
length)
seSerotonin <- sdSerotonin / sqrt(nSerotonin)
Error bars
Draw the strip chart and then add the error bars.
\[ \bar{Y} \pm SE_{\bar{Y}} \]
offsetAmount <- 0.2
stripchart(serotoninLevel ~ treatmentTime,
data = locustData,
method = "jitter",
vertical = TRUE)
segments(1:3 + offsetAmount,
meanSerotonin - seSerotonin,
1:3 + offsetAmount,
meanSerotonin + seSerotonin)
points(meanSerotonin ~ c(c(1,2,3) + offsetAmount),
pch = 16,
cex = 1.2)
Error bars
Draw the strip chart and then add the error bars.
\[ \bar{Y} \pm SE_{\bar{Y}} \]
Error bars can mean different things!!!
Different error bars!!! \[ \bar{Y} \pm sd \\ \bar{Y} \pm SE_{\bar{Y}} \\ \bar{Y} \pm 2\times SE_{\bar{Y}} \]