Question

In an effort to characterize the New Guinea crocodile (Crocodylus novaeguineae), measurements were taken of the dorsal cranial length (mm) (the length of the skull from the tip of the nose to the back of the cranial cap, denoted DCL) and the total length (cm) (denoted TL) of 50 harvested adult males. (Data on next page.)

1. Construct a histogram and a boxplot for each of the variables DCL and TL. Comment on the symmetry of the distribution of each variable.

2. Construct a scatterplot with DCL on the vertical axis and TL on the horizontal axis. Based on this plot, what can be said about the relationship between DCL and TL (i.e. if you vary DCL, what happens to TL)?

Data

TL           DCL           Observation

130           169           1

102           154           2

126           160           3

230           290           4

115           151           5

150           209           6

259           344           7

130           183           8

110           153           9

130           183           10

185           237           11

215           288           12

129           187           13

149           189           14

156           203           15

100           143           16

224           294           17

234           318           18

162           229           19

217           299           20

206           283           21

144           198           22

146           203           23

166           229           24

203           275           25

205           266           26

252           350           27

238           318           28

250           330           29

255           351           30

120           169           31

250           332           32

238           307           33

157           205           34

159           216           35

202           261           36

177           237           37

221           288           38

224           294           39

167           232           40

240           316           41

207           268           42

192           242           43

180           248           44

165           226           45

197           267           46

113           162           47

131           183           48

162           234           49

246           310           50

Answer 1

Following is the raw data for our analysis given in a tabular form:

TL	DCL
130	169
102	154
126	160
230	290
115	151
150	209
259	344
130	183
110	153
130	183
185	237
215	288
129	187
149	189
156	203
100	143
224	294
234	318
162	229
217	299
206	283
144	198
146	203
166	229
203	275
205	266
252	350
238	318
250	330
255	351
120	169
250	332
238	307
157	205
159	216
202	261
177	237
221	288
224	294
167	232
240	316
207	268
192	242
180	248
165	226
197	267
113	162
131	183
162	234
246	310

Note : All the required plots namely, the histogram, the boxplot and the scatterplot are constructed using Python's Seaborn library in order to complete the project within the time bounds . The steps for manually plotting the same shall be mentioned for a refference:

A. Following steps must be followed in order to construct a histogram for a given data atteribute :

1- Find the smallest and the largest number in the given data.

2- Find the range of the data by subtracting the largest number from the smallest number.

3- Now, we need to find the width of our class using the range. There is no hard-and-fast rule for this.So, we do this intituively by deviding the range by 5 for small ranges any by 10 in large ranges e.g, in our case for the atteribute TL, we devide the range by 10 and round off to obtain a class of width 16 units.

4- Make a frequency table having these two columns:

a) Intervals of the class width ,starting from the minimum to the maximum as per the class width and

b) The number of data points existing in that particular interval.

5- Draw perpendicular lines, the x- and y-axes. Place the frequencies on y-axis and the lower value of respective intervals on the x-axis.with the atteribute name.

6- Draw the bars whose width is from the lower value of first interval to the lower value of the second interval, and so on.

B. Following steps must be followed in order to construct a boxplot for a given data atteribute :

1-Find the min and max values of the atteribute.

2- Calculate the following 3 descriptive quantities of the atteribute:

a) median (Q2)

Formula:

- Sort the data in ascending order.

- If the total number of data-points is an odd number, the median is the {(n+1)/2}th observation .

- If the total number of data-points is an even number, the median is the average of the  (n/2)th and the {(n+1)/2}th observations.

b) 25-th percentile (Q1)

c) 75-th percentile (Q3)

3- Draw a horizontal rectangular box and lets its first edge be Q1 and the second edge be Q3.

4- Devide the rectangle by drawing a vertical line inside it, the median, whose distance from Q1 and Q3 scaled accordin to its magnitude relative to them. Call it Q2

5- Extend horizontal lines on both the sides of scale relative to min and the max valies on the Q1 and Q3, respectively. Cal them min and max

C. Following steps must be followed in order to construct a scatterplot for a given data atteributes :

1- Draw perpendicular lines, the x- and y-axes. Place the first atteribute name on y-axis and the second atteribute name on the x-axis.

2. Make dots corresponding to the adjecent observation pairs with respect to both the axes.

1. Construct a histogram and a boxplot for each of the variables DCL and TL. Comment on the symmetry of the distribution of each variable.

ANS)

DCL atteribute uni-variate analysis:

a. Histogram:

b. Boxplot

OBSERVATIONS:

1- The data is about-normally distributed with a marginal right-skew.

2- Since the skew is minimal, no extreme values are present at all.

3 - Since more than one peaks are appearent, the distribution is bi-modal, hinting in presence two independent groups or clusters.

4- Since the distribution is not having a shark peak, it is a plati-kurtic (-ve kurtosis) distribution citing the fact that the values are not very centered towatds the mean but have a healthy spread.

5- The range of values being 200, is significant citing high variation on length of species.

TL atteribute uni-variate analysis:

a. Histogram:

b. Boxplot

OBSERVATIONS:

1- The data is about-normally distributed with a very marginal right-skew.

2- Since the skew is minimal, no extreme values are present at all in this atteribute as well

3 - Since more than one peaks are appearent, the distribution is bi-modal, high chances of presence two independent groups .

4- It is a plati-kurtic (-ve kurtosis) distribution citing the fact tthat it has a healthy spread, and not centered towards the mean.

5- The range of values being 160, is significant citing high variation on length but less varience than DCL atteribute.

Conclusion can be made that both these atteributes are having roughly identical normal distribition , with to hidden groups.

2. Construct a scatterplot with DCL on the vertical axis and TL on the horizontal axis. Based on this plot, what can be said about the relationship between DCL and TL (i.e. if you vary DCL, what happens to TL)?

OBSERVATIONS

The scatterplot between the variables TL and DCL suggest a very high level of correlation. I.e, there is a positive dependecy between the both quantities. So, with a given increase in the TL, the DCL will probably increase as well and vice-versa.