Means and Uncertainty

# Means and Uncertainty
### Derek Ogle, May 2020

---

# Background

- The Great Lakes Environmental Research Laboratory (GLERL) of the National Oceanic and Atmospheric Administration (NOAA) uses satellites to annually monitor ice coverage on the Great Lakes.
  - Maximum ice coverage recorded for all five lakes since 1972.
  - Data originally accessed from [the GLERL website](https://www.glerl.noaa.gov/data/ice/#historical) and are [in this file](https://raw.githubusercontent.com/droglenc/NCData/master/GreatLakesIce.csv").

.center[
<img src="https://derekogle.com/NCGraphing/modules/zimgs/GreatLakesIce.jpg" width="400px" />
]

---

# Background

- Loaded data into `gli` data.frame.
  - Ordered lakes from west to east (use of `levels=` in `factor()`).
  - Created a `decade` variable.

```r
#!# Set to your own working directory and have just your filename below.
gli <- read.csv("https://raw.githubusercontent.com/droglenc/NCData/master/GreatLakesIce.csv") %>%
 mutate(lake=factor(lake,levels=c("Superior","Michigan","Huron","Erie","Ontario")),
 decade=floor(year/10)*10,
 decade=factor(paste0(decade,"s"),levels=c("1970s","1980s","1990s","2000s","2010s")))
```

---

# Background

```r
head(gli)
```

```
      lake year   date max.cover decade
1 Superior 1973 27-Feb      69.8  1970s
2 Superior 1974 27-Feb      73.8  1970s
3 Superior 1975 13-Feb      64.9  1970s
4 Superior 1976 16-Mar      49.9  1970s
5 Superior 1977 16-Feb      96.0  1970s
6 Superior 1978  1-Mar      92.5  1970s
```

- Variables used are:
  - **lake**: Great lake.
  - **year**: Year of ice season.
  - **max.cover**: Maximum percent of lake covered with ice.
  - **decade**: Decade of ice season.

--
- Color palette used is:

```r
clrs <- c("Superior"="#000000","Michigan"="#E69F00","Huron"="#56B4E9",
 "Erie"="#009E73","Ontario"="#CC79A7")
```

---

# Exploratory Data Visualizations

---

```r
*head(gli)
```
]]
.column[.content[

```r
head(gli)

*ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake)) +  
* geom_line(size=0.4,alpha=0.75)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_3_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake)) +  
  geom_line(size=0.4,alpha=0.75) +  
* scale_x_continuous(name="Annual Ice Season")
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_4_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake)) +  
  geom_line(size=0.4,alpha=0.75) +  
  scale_x_continuous(name="Annual Ice Season") +  
* scale_y_continuous(name="Maximum Ice Coverage (%)",
*                    limits=c(0,100),expand=expansion(mult=0.01))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_5_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake)) +  
  geom_line(size=0.4,alpha=0.75) +  
  scale_x_continuous(name="Annual Ice Season") +  
  scale_y_continuous(name="Maximum Ice Coverage (%)",  
                     limits=c(0,100),expand=expansion(mult=0.01)) +  
* scale_color_manual(name="Lake",values=clrs)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_6_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=year,y=max.cover,color=lake)) +  
  geom_line(size=0.4,alpha=0.75) +  
  scale_x_continuous(name="Annual Ice Season") +  
  scale_y_continuous(name="Maximum Ice Coverage (%)",  
                     limits=c(0,100),expand=expansion(mult=0.01)) +  
  scale_color_manual(name="Lake",values=clrs) +  
* theme_bw()
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv1_auto_7_output-1.png" width="100%" />
]]

---

# A Peek into a Future Module

---

---

```r
*head(gli)
```
]]
.column[.content[

```r
head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
* geom_violin(color="gray50",fill="gray50",alpha=0.75,trim=FALSE)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_3_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  geom_violin(color="gray50",fill="gray50",alpha=0.75,trim=FALSE) +  
* geom_jitter(width=0.07,size=0.5,color="black",alpha=0.5)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_4_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_5_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  geom_violin(color="gray50",fill="gray50",alpha=0.75,trim=FALSE) +  
  geom_jitter(width=0.07,size=0.5,color="black",alpha=0.5) +  
  scale_x_discrete(name="Great Lake") +  
* scale_y_continuous(name="Maximum Ice Coverage (%)",
*                    limits=c(0,100),expand=expansion(mult=0.01))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_6_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  geom_violin(color="gray50",fill="gray50",alpha=0.75,trim=FALSE) +  
  geom_jitter(width=0.07,size=0.5,color="black",alpha=0.5) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Maximum Ice Coverage (%)",  
                     limits=c(0,100),expand=expansion(mult=0.01)) +  
* theme_bw()
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/edv2_auto_7_output-1.png" width="100%" />
]]

---

# Means Plots

---

## Using `stat_summary()`

- Computes a summary of data and applies that to a `geom_`.

&nbsp;

--
- Is flexible -- we will use in two different ways.

&nbsp;

--
- First way is to compute a summary with a function in `fun=`.
  - For example, `fun=mean` will compute the mean of "Y" for each "X".

&nbsp;

--
- The summary is then plotted according to a specified `geom_` in `geom=`.
  - For example, `geom="bar"` or `geom="point"`.

---

```r
*head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar1_user_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
* stat_summary(fun=mean,geom="bar",
*              fill="gray30",alpha=0.75)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar1_user_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="bar",  
               fill="gray30",alpha=0.75) +  
* scale_x_discrete(name="Great Lake") +
* scale_y_continuous(name="Mean Maximum Ice Coverage (%)",
*                    limits=c(0,100),expand=expansion(mult=c(0,0.01))) +
* theme_bw() +
* theme(panel.grid.major.x=element_blank())
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar1_user_3_output-1.png" width="100%" />
]]

---

```r
*head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts1_user_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
* stat_summary(fun=mean,geom="point")
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts1_user_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="point") +  
* scale_x_discrete(name="Great Lake") +
* scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +
* theme_bw() +
* theme(panel.grid.major.x=element_blank())
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts1_user_3_output-1.png" width="100%" />
]]

---

# Means Plots with Uncertainty

---

## Uncertainty?

- Primarily related to variability.

&nbsp;

--
- Standard deviation (SD) measures variability of individuals.

&nbsp;

--
- Standard error (SE) measures variability of a statistic (from sample to sample).
  - SE=SD/sqrt(n)

&nbsp;

--
- Margin-of-error (ME) measure uncertainty in our inference for (estimate of) an unknown parameter.
 - ME = t\*SE, where t\* is from a t distribution with n-1 df

---

## Uncertainty

- Generally visualized as intervals around a point estimate (e.g., a mean).

&nbsp;

--
- mean &pm; SD
  - Shows variability of individuals around the mean.

&nbsp;

--
- mean &pm; SE
  - Shows variability of a statistic (from sample to sample).

&nbsp;

--
- mean &pm; ME
  - Shows an interval for which we are 95% confident that the unknown parameter will fall.

---

## Using `stat_summary()`

- Computes a summary of the data and applies that to a `geom_`.
- Is flexible -- we will use in two different ways.

&nbsp;

--
- Second way is to compute a data.frame with "center", minimum, and maximum values with a function in `fun.data=`.

--
- Built-in functions that can be used in `fun.data=`.
  - `mean_se`: computes mean ("center") and plus/minus a SE.
  - `mean_sdl`: computes mean and plus/minus **two** SDs.
  - `mean_cl_normal`: computes mean and confidence interval from normal distribution theory.
  - `mean_cl_boot`: computes mean and confidence interval from bootstrapping.

---

## Intervals added with `geom_errorbar()`

- New `aes()`thetics.
  - `ymin=`: the lower value for the error bar.
  - `ymax=`: the upper value for the error bar.

--
- These new `aes()`thetics are automatic from the built-in functions used in `stat_summary()`.

&nbsp;

--
- Control width of end caps on error bar with `width=`.

---

```r
*head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar2_user_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
* stat_summary(fun=mean,geom="bar",fill="gray30",alpha=0.75)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar2_user_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="bar",fill="gray30",alpha=0.75) +  
* stat_summary(fun.data=mean_cl_normal,geom="errorbar",
*              width=0.25)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar2_user_3_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="bar",fill="gray30",alpha=0.75) +  
  stat_summary(fun.data=mean_cl_normal,geom="errorbar",  
               width=0.25) +  
* scale_x_discrete(name="Great Lake") +
* scale_y_continuous(name="Mean Maximum Ice Coverage (%)",
*                    limits=c(0,100),expand=expansion(mult=c(0,0.01))) +
* theme_bw() +
* theme(panel.grid.major.x=element_blank())
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnsbar2_user_4_output-1.png" width="100%" />
]]

---

```r
*head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover))
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts2_user_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
* stat_summary(fun=mean,geom="point",)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts2_user_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="point",) +  
* stat_summary(fun.data=mean_cl_normal,geom="errorbar",width=0.25)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts2_user_3_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(fun=mean,geom="point",) +  
  stat_summary(fun.data=mean_cl_normal,geom="errorbar",width=0.25) +  
* scale_x_discrete(name="Great Lake") +
* scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +
* theme_bw() +
* theme(panel.grid.major.x=element_blank())
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts2_user_4_output-1.png" width="100%" />
]]

---

## Intervals added with `geom_pointrange()`

- Similar to `geom_errorbar()` but also plots the "center" value.

&nbsp;

--
- No end caps on interval, thus no `width=` argument.

--
- `size=`: controls the size of the range line AND point.
- `fatten=`: subsequently controls the size of the center point (as a multiplier).

---

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",  
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts3_non_seq_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",  
*   size=1,
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts3_non_seq_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",  
    size=1,  
*   fatten=2
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts3_non_seq_3_output-1.png" width="100%" />
]]

---

# Show Means/Intervals with Data

---

```r
*head(gli)

*ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +
* geom_violin(color="gray30",fill="gray50",alpha=0.1,trim=FALSE)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnvio_user_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  geom_violin(color="gray30",fill="gray50",alpha=0.1,trim=FALSE) +  
* geom_jitter(data=gli,mapping=aes(x=lake,y=max.cover),
*             width=0.07,size=0.5,color="black",alpha=0.25)
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnvio_user_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

<img src="Lecture_GreatLakesIce_files/figure-html/mnvio_user_3_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(x=lake,y=max.cover)) +  
  geom_violin(color="gray30",fill="gray50",alpha=0.1,trim=FALSE) +  
  geom_jitter(data=gli,mapping=aes(x=lake,y=max.cover),  
              width=0.07,size=0.5,color="black",alpha=0.25) +  
  stat_summary(fun.data=mean_cl_normal,geom="pointrange",  
               size=0.75,fatten=1,pch=3,color="red") +  
* scale_x_discrete(name="Great Lake") +
* scale_y_continuous(name="Maximum Ice Coverage (%)",
*                    limits=c(0,100),expand=expansion(mult=0.01)) +
* theme_bw()
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnvio_user_4_output-1.png" width="100%" />
]]

---

# Means Plots with Multiple Groupings

---

```r
head(gli)

ggplot(data=gli,mapping=aes(  
    x=lake,y=max.cover,  
  )) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",size=0.5,fatten=1,  
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts4_non_seq_1_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(  
    x=lake,y=max.cover,  
*   color=decade
  )) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",size=0.5,fatten=1,  
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts4_non_seq_2_output-1.png" width="100%" />
]]
---
class: split-50
count: false

```r
head(gli)

ggplot(data=gli,mapping=aes(  
    x=lake,y=max.cover,  
    color=decade  
  )) +  
  stat_summary(  
    fun.data=mean_cl_normal,geom="pointrange",size=0.5,fatten=1,  
*   position=position_dodge(width=0.5)
  ) +  
  scale_x_discrete(name="Great Lake") +  
  scale_y_continuous(name="Mean Maximum Ice Coverage (%)") +  
  theme_bw() +  
  theme(panel.grid.major.x=element_blank())  
```
]]
.column[.content[

<img src="Lecture_GreatLakesIce_files/figure-html/mnspts4_non_seq_3_output-1.png" width="100%" />
]]

---

# Means Plots
### From Summarized Results

---

```r
*sum1 <- gli
 
sum1
```
]]
.column[.content[

```
        lake year   date max.cover decade
1   Superior 1973 27-Feb     69.80  1970s
2   Superior 1974 27-Feb     73.80  1970s
3   Superior 1975 13-Feb     64.90  1970s
4   Superior 1976 16-Mar     49.90  1970s
5   Superior 1977 16-Feb     96.00  1970s
6   Superior 1978  1-Mar     92.50  1970s
7   Superior 1979 25-Feb     97.10  1970s
8   Superior 1980  5-Mar     77.20  1980s
9   Superior 1981 11-Feb     84.70  1980s
10  Superior 1982  9-Mar     85.30  1980s
11  Superior 1983  7-Feb     20.20  1980s
12  Superior 1984  7-Mar     88.20  1980s
13  Superior 1985  2-Mar     81.90  1980s
14  Superior 1986 22-Feb     90.70  1980s
15  Superior 1987 25-Jan     14.50  1980s
16  Superior 1988 15-Feb     67.30  1980s
17  Superior 1989  8-Mar     82.30  1980s
18  Superior 1990  5-Mar     78.10  1990s
19  Superior 1991  4-Mar     89.60  1990s
20  Superior 1992 23-Mar     73.30  1990s
21  Superior 1993  1-Mar     77.20  1990s
22  Superior 1994  9-Feb     96.10  1990s
23  Superior 1995  3-Mar     28.80  1990s
24  Superior 1996  8-Mar    100.00  1990s
25  Superior 1997 23-Mar     89.00  1990s
26  Superior 1998 16-Jan     11.10  1990s
27  Superior 1999 23-Feb     18.30  1990s
28  Superior 2000 18-Feb     33.50  2000s
29  Superior 2001 12-Mar     46.70  2000s
30  Superior 2002 17-Mar     10.30  2000s
31  Superior 2003  4-Mar     95.50  2000s
32  Superior 2004 19-Feb     52.20  2000s
33  Superior 2005 17-Mar     54.50  2000s
34  Superior 2006  2-Mar     16.66  2000s
35  Superior 2007  8-Mar     51.83  2000s
36  Superior 2008 10-Mar     62.70  2000s
37  Superior 2009  2-Mar     93.63  2000s
38  Superior 2010 22-Feb     27.34  2010s
39  Superior 2011  2-Apr     33.64  2010s
40  Superior 2012 22-Jan      8.22  2010s
41  Superior 2013 17-Feb     38.62  2010s
42  Superior 2014  3-Mar     95.82  2010s
43  Superior 2015 28-Feb     95.70  2010s
44  Superior 2016 14-Feb     22.69  2010s
45  Superior 2017  4-Mar     18.70  2010s
46  Superior 2018 11-Feb     77.17  2010s
47  Superior 2019  8-Mar     94.91  2010s
48  Michigan 1973 27-Feb     33.00  1970s
49  Michigan 1974 21-Feb     39.40  1970s
50  Michigan 1975 29-Jan     28.10  1970s
51  Michigan 1976  3-Feb     29.50  1970s
52  Michigan 1977  9-Feb     93.10  1970s
53  Michigan 1978 22-Feb     66.60  1970s
54  Michigan 1979 25-Feb     92.30  1970s
55  Michigan 1980 20-Feb     38.60  1980s
56  Michigan 1981 21-Jan     53.80  1980s
57  Michigan 1982  9-Feb     60.20  1980s
58  Michigan 1983  8-Feb     23.60  1980s
59  Michigan 1984  8-Feb     43.30  1980s
60  Michigan 1985 23-Feb     41.30  1980s
61  Michigan 1986 23-Feb     66.80  1980s
62  Michigan 1987 21-Feb     20.00  1980s
63  Michigan 1988  7-Feb     32.70  1980s
64  Michigan 1989 19-Mar     30.90  1980s
65  Michigan 1990 14-Jan     32.40  1990s
66  Michigan 1991 30-Jan     21.50  1990s
67  Michigan 1992  3-Feb     32.80  1990s
68  Michigan 1993 28-Feb     32.20  1990s
69  Michigan 1994 14-Feb     82.70  1990s
70  Michigan 1995 13-Feb     21.60  1990s
71  Michigan 1996  7-Feb     75.00  1990s
72  Michigan 1997  5-Feb     37.80  1990s
73  Michigan 1998  6-Feb     15.10  1990s
74  Michigan 1999 15-Jan     23.00  1990s
75  Michigan 2000 28-Jan     27.20  2000s
76  Michigan 2001 22-Feb     29.50  2000s
77  Michigan 2002  7-Mar     12.40  2000s
78  Michigan 2003 10-Mar     48.00  2000s
79  Michigan 2004  5-Feb     36.40  2000s
80  Michigan 2005 24-Jan     29.40  2000s
81  Michigan 2006 20-Feb     16.15  2000s
82  Michigan 2007  8-Mar     37.19  2000s
83  Michigan 2008 21-Feb     33.51  2000s
84  Michigan 2009 29-Jan     52.27  2000s
85  Michigan 2010  8-Feb     23.46  2010s
86  Michigan 2011 11-Feb     29.35  2010s
87  Michigan 2012 21-Jan     16.72  2010s
88  Michigan 2013 18-Feb     24.38  2010s
89  Michigan 2014  8-Mar     93.08  2010s
90  Michigan 2015 27-Feb     72.90  2010s
91  Michigan 2016 14-Feb     26.75  2010s
92  Michigan 2017 18-Jan     18.20  2010s
93  Michigan 2018 11-Feb     51.32  2010s
94  Michigan 2019  8-Mar     55.83  2010s
95     Huron 1973 26-Feb     66.70  1970s
96     Huron 1974 27-Feb     61.40  1970s
97     Huron 1975 13-Feb     55.10  1970s
98     Huron 1976 28-Jan     52.50  1970s
99     Huron 1977  9-Feb     95.00  1970s
100    Huron 1978  1-Mar     95.80  1970s
101    Huron 1979 19-Feb     96.40  1970s
102    Huron 1980  5-Mar     70.30  1980s
103    Huron 1981  4-Feb     93.80  1980s
104    Huron 1982  9-Mar     93.30  1980s
105    Huron 1983  1-Feb     31.80  1980s
106    Huron 1984 14-Mar     77.60  1980s
107    Huron 1985 23-Feb     87.40  1980s
108    Huron 1986 23-Feb     73.50  1980s
109    Huron 1987 15-Feb     43.50  1980s
110    Huron 1988  6-Mar     60.10  1980s
111    Huron 1989 19-Mar     76.30  1980s
112    Huron 1990  7-Mar     73.80  1990s
113    Huron 1991  1-Feb     43.80  1990s
114    Huron 1992 23-Mar     69.90  1990s
115    Huron 1993 28-Feb     78.30  1990s
116    Huron 1994  2-Mar     96.90  1990s
117    Huron 1995  5-Mar     41.70  1990s
118    Huron 1996 19-Feb     98.20  1990s
119    Huron 1997 23-Feb     65.30  1990s
120    Huron 1998  6-Feb     28.60  1990s
121    Huron 1999  9-Mar     36.00  1990s
122    Huron 2000 18-Feb     41.90  2000s
123    Huron 2001 19-Feb     45.70  2000s
124    Huron 2002  4-Mar     26.10  2000s
125    Huron 2003  8-Mar     96.20  2000s
126    Huron 2004 19-Feb     64.50  2000s
127    Huron 2005  9-Mar     58.90  2000s
128    Huron 2006  2-Mar     32.17  2000s
129    Huron 2007  8-Mar     71.39  2000s
130    Huron 2008 17-Mar     59.51  2000s
131    Huron 2009  2-Mar     85.11  2000s
132    Huron 2010 18-Feb     36.95  2010s
133    Huron 2011 24-Feb     63.79  2010s
134    Huron 2012  5-Mar     23.06  2010s
135    Huron 2013 18-Feb     47.75  2010s
136    Huron 2014  6-Mar     96.10  2010s
137    Huron 2015  6-Mar     96.40  2010s
138    Huron 2016 14-Feb     47.98  2010s
139    Huron 2017 14-Mar     35.35  2010s
140    Huron 2018 11-Feb     81.39  2010s
141    Huron 2019  9-Mar     95.70  2010s
142     Erie 1973 27-Feb     95.70  1970s
143     Erie 1974 21-Feb     88.50  1970s
144     Erie 1975 13-Feb     80.10  1970s
145     Erie 1976  8-Feb     95.40  1970s
146     Erie 1977  8-Feb     99.80  1970s
147     Erie 1978  8-Feb    100.00  1970s
148     Erie 1979 12-Feb    100.00  1970s
149     Erie 1980  5-Mar     93.40  1980s
150     Erie 1981 14-Jan     96.00  1980s
151     Erie 1982 15-Feb     99.10  1980s
152     Erie 1983 14-Feb     40.80  1980s
153     Erie 1984 18-Jan     95.70  1980s
154     Erie 1985  9-Feb     96.00  1980s
155     Erie 1986  1-Feb     95.50  1980s
156     Erie 1987 14-Feb     88.00  1980s
157     Erie 1988  7-Feb     91.50  1980s
158     Erie 1989 17-Feb     91.60  1980s
159     Erie 1990  9-Mar     72.80  1990s
160     Erie 1991  1-Feb     35.10  1990s
161     Erie 1992 16-Feb     89.80  1990s
162     Erie 1993 24-Feb     94.30  1990s
163     Erie 1994  9-Feb     96.70  1990s
164     Erie 1995 13-Feb     94.00  1990s
165     Erie 1996  5-Feb    100.00  1990s
166     Erie 1997  2-Feb     99.60  1990s
167     Erie 1998 23-Jan      5.40  1990s
168     Erie 1999 15-Jan     74.80  1990s
169     Erie 2000  1-Feb     90.70  2000s
170     Erie 2001 12-Feb     94.00  2000s
171     Erie 2002  3-Jan     14.40  2000s
172     Erie 2003 27-Jan     95.70  2000s
173     Erie 2004  2-Feb     95.40  2000s
174     Erie 2005  7-Feb     93.00  2000s
175     Erie 2006  2-Mar     21.88  2000s
176     Erie 2007 19-Feb     95.77  2000s
177     Erie 2008 10-Mar     93.38  2000s
178     Erie 2009  5-Feb     95.55  2000s
179     Erie 2010  8-Feb     93.13  2010s
180     Erie 2011  1-Feb     95.85  2010s
181     Erie 2012 21-Jan     13.87  2010s
182     Erie 2013 22-Feb     83.72  2010s
183     Erie 2014  6-Mar     96.08  2010s
184     Erie 2015 18-Feb     98.10  2010s
185     Erie 2016 15-Feb     78.73  2010s
186     Erie 2017  8-Feb     35.51  2010s
187     Erie 2018  9-Feb     95.12  2010s
188     Erie 2019  2-Mar     94.28  2010s
189  Ontario 1973 20-Feb     62.60  1970s
190  Ontario 1974 10-Jan     24.50  1970s
191  Ontario 1975 13-Feb     14.90  1970s
192  Ontario 1976  3-Feb     15.20  1970s
193  Ontario 1977 17-Feb     46.70  1970s
194  Ontario 1978 15-Feb     57.70  1970s
195  Ontario 1979 19-Feb     86.20  1970s
196  Ontario 1980 13-Feb     30.60  1980s
197  Ontario 1981  7-Jan     47.60  1980s
198  Ontario 1982 25-Jan     50.70  1980s
199  Ontario 1983  8-Feb     12.50  1980s
200  Ontario 1984 18-Jan     40.70  1980s
201  Ontario 1985  9-Feb     38.20  1980s
202  Ontario 1986 28-Feb     43.70  1980s
203  Ontario 1987 14-Feb      8.40  1980s
204  Ontario 1988 10-Jan     21.10  1980s
205  Ontario 1989 13-Mar     15.50  1980s
206  Ontario 1990 28-Feb     29.50  1990s
207  Ontario 1991  2-Feb     11.60  1990s
208  Ontario 1992 16-Feb     17.50  1990s
209  Ontario 1993 28-Feb     29.00  1990s
210  Ontario 1994 11-Feb     55.70  1990s
211  Ontario 1995 13-Feb     18.80  1990s
212  Ontario 1996 16-Feb     45.10  1990s
213  Ontario 1997 23-Feb     25.60  1990s
214  Ontario 1998 30-Jan      6.20  1990s
215  Ontario 1999  9-Mar     17.90  1990s
216  Ontario 2000  1-Feb     22.30  2000s
217  Ontario 2001 15-Feb     17.90  2000s
218  Ontario 2002 10-Jan      4.00  2000s
219  Ontario 2003 27-Feb     49.60  2000s
220  Ontario 2004 29-Jan     38.50  2000s
221  Ontario 2005 31-Jan     37.80  2000s
222  Ontario 2006  2-Mar     14.25  2000s
223  Ontario 2007  8-Mar     23.77  2000s
224  Ontario 2008 10-Mar     13.97  2000s
225  Ontario 2009  5-Feb     25.90  2000s
226  Ontario 2010  8-Feb     13.10  2010s
227  Ontario 2011 24-Feb     31.09  2010s
228  Ontario 2012 17-Feb      1.86  2010s
229  Ontario 2013 23-Feb     15.58  2010s
230  Ontario 2014  6-Mar     60.84  2010s
231  Ontario 2015 18-Feb     82.40  2010s
232  Ontario 2016 14-Feb     23.46  2010s
233  Ontario 2017 16-Mar      5.75  2010s
234  Ontario 2018 17-Jan     24.67  2010s
235  Ontario 2019  1-Mar     39.78  2010s
```
]]
---
class: split-50
count: false

```r
sum1 <- gli %>% 
* group_by(lake)
 
sum1
```
]]
.column[.content[

```
# A tibble: 235 x 5
# Groups: lake [5]
 lake year date max.cover decade
 <fct> <int> <fct> <dbl> <fct> 
 1 Superior 1973 27-Feb 69.8 1970s 
 2 Superior 1974 27-Feb 73.8 1970s 
 3 Superior 1975 13-Feb 64.9 1970s 
 4 Superior 1976 16-Mar 49.9 1970s 
 5 Superior 1977 16-Feb 96 1970s 
 6 Superior 1978 1-Mar 92.5 1970s 
 7 Superior 1979 25-Feb 97.1 1970s 
 8 Superior 1980 5-Mar 77.2 1980s 
 9 Superior 1981 11-Feb 84.7 1980s 
10 Superior 1982 9-Mar 85.3 1980s 
# ... with 225 more rows
```
]]
---
class: split-50
count: false

```r
sum1 <- gli %>% 
 group_by(lake) %>% 
* summarize(n=n(),
* mean=mean(max.cover,na.rm=TRUE))
 
sum1
```
]]
.column[.content[

```
# A tibble: 5 x 3
 lake n mean
 <fct> <int> <dbl>
1 Superior 47 62.3
2 Michigan 47 40.5
3 Huron 47 65.3
4 Erie 47 82.6
5 Ontario 47 30.2
```
]]

---

# Means Plots

### Plot Summarized Means

---

```r
*sum1

*ggplot(data=sum1,mapping=aes(x=lake,y=mean)) +
* geom_bar(stat="identity",fill="gray30",alpha=0.75)
```
]]
.column[.content[