Activity 2. Analysis of Leaves Length-Width Relationship

Using Basic Statistical Tools

Renzo Val Agapito, Miguel Luis Chua, Rayniel Frio and Malaya Turzar

Biology Student, Department of Biology, College of Science, Polytechnic University of the Philippines

Abstract

Data analysis in ecology is drawn from a jumbled mass of measurements. Statistical

inference uses logical and repeatable methods to extract information from noisy data. It also

helps us in arranging the random collection of numerical information into a comprehensive table.

Frequency distribution table (FDT) summarized the collated data and linear regression and

correlation analysis defined the relationship of leaves length and width. With the use of different

statistical tools and methods, it was determined that the leaves generally measures about 9-16

mm. Linear regression and correlation proved that leaves length and width are strongly

correlated and are directly proportional to each other.

Keywords: Statistics, frequency distribution table, linear regression, linear correlation.

Introduction

Plant leaves are the major site of

light interception and consequently where

metabolic processes occur including

transpiration (Goudriaan and Van Laar,

1994). Accurate and simple measurements

of leaf area (LA) of a crop are essential to

understand the interaction between plant

growth and environment since it is an

indicator of plant growth and productivity. It

is also a determinant factor in mechanisms

such as light interception, photosynthetic

efficiency, evapotranspiration, energy

exchange and responses to fertilizers and

(De Jesus 2001; Blanco and Folegatti 2005;

Demirsoy 2004).

Data analysis in ecology is drawn

from a jumbled mass of measurements.

Statistical inference uses logical and

repeatable methods to extract information

from noisy data. It also helps us in arranging

the random collection of numerical

information into a comprehensive table – a

frequency distribution table. It displays the

frequency of various outcomes in a sample

and it summarizes the distribution of values

in the sample. Important statistics must be

included in the table including classes and

frequency.

Various combinations of

measurements and various equations relating

length and width to area have been

developed for several such horticultural

crops as cucumber (Cho et al., 2007;

Robbins and Pharr, 1987). Linear

measurements are more advantageous than

any other methods in measuring the leaf

estimated surface area if one of the

mathematical relationships between leaf area

and one or more dimensions of leaf are

clarified (Beerling and Fry, 1990; Villegas

et al., 1981).

Scientific studies often desired to

investigate relationships between variables

in one linear measurement. In this study, the

length-width ratios of leaf samples were

investigated and computed the linear

correlation, and measure the individual stats

of leaves length and width. In presenting

relationships, patterns and trends, the usage

of graphs or diagrams (i.e. Frequency

Distribution Table, Scatter Diagram) helps

in interpreting the significance of the results.

Methodology

Sample Collection and Measurement

Leaflets of Caesalpinia pulcherrima

were collected from Brgy. Pinyahan, Quezon

City. 100 leaflets were measured with regards

to its length and width using a ruler with

millimeter calibration. The greatest width of

leaflets were measured to the nearest

millimeter same with the length whose

measured from the tip of the apex to its base.

Data Analysis

Data gathered were arranged using

Stem and Leaf Diagram and analyzed by

means of Frequency Distribution Table

(FDT). Scatter Diagram were formulated to

determine the relationship of length and

width. Statistics were used for the mean,

variance, standard deviation, variance of the

mean and standard error was also computed.

Mean (𝑥) = ∑ 𝑥

𝑛

Variance 𝑠2 =

Standard Deviation 𝑠 = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

Variance of the Mean = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

𝑛

1

2

n

n

x

Standard Error =𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

√𝑛

Presentation of Data

The data and observations gathered

were presented with the use of regression

equation given by the formula 𝑌 = 𝑎 + 𝑏𝑥

and linear correlation.

The slope, (b) was computed using:

𝑏 = ∑ 𝑥𝑦 − 𝐶𝑇𝑥𝑦

∑ 2𝑥 − 𝐶𝑇𝑥

Computation of the estimated

intercept (a), is as follows:

𝑎 = 𝑦 − 𝑏𝑥

= ∑ 𝑦

𝑛− 𝑏

∑ 𝑥

𝑛

The correlation equation, (r) is

computed by:

𝑟 = ∑ 𝑥𝑦 − 𝐶𝑇𝑥𝑦

√(∑ 2𝑥 − 𝐶𝑇𝑥) − (∑ 2 − 𝐶𝑇𝑦)𝑦

The value computed for correlation

coefficient is interpreted using the Pearson’s

r values table as shown below:

Value Interpretation

0.00-0.19 Very weak correlation

0.20-0.39 Weak correlation

0.40-0.69 Modest correlation

0.70-0.89 Strong correlation

0.90-1.00 Very strong

correlation

Results and Discussions

Leaf Description

Caesalpinia pulcherrima L. has

opposite phyllotaxy, bipinnately compound

leaf with an entire margin usually oblong or

elliptic in shape. The leaflets are an

evergreen usually in 6-10 pairs measuring

15-25 mm long and 10-15 mm broad.

Measure of Central Location

Table 1. Measure of Central Location of Caesalpinia

pulcherrima L. Leaf Length and Width

Mean (�̅�) Median (�̃�) Mode (�̌�)

Width 9.03 9 9

Length 15.92 16 16

Table 1 depicted the most

representative data of the sample, 9.03 for

the width and 15.92 in length.

Sample Mean and Variance

Table 2. Sample Mean and Variance of Caesalpinia

pulcherrima L. Leaf Length and Width

Width (X) Length (Y)

Mean 9.03 15.92

Variance 1.95 6.07

Std.

Deviation

1.40 2.46

Variance of

the mean

0.02 0.06

Standard error 0.14 0.25

Table 2 showed the difference of

variance from the mean. Both width and

length presented small variance with 1.95

and 6.07 respectively. Small variance

indicates that the data points tend to be very

close to the mean or the expected value

(Loeve, M. 1977). Standard deviation also

showed low value with 1.40 for the width

and 2.46 for length indicating that data

points tend to be very close to the mean

(Loeve, M. 1977). The standard error shows

how the individual data varies to the mean

(Loeve, M. 1977). The standard error for

width and length is 0.14 and 0.25

respectively.

Frequency Distribution

Frequency distribution summarizes

the number of occurrences of values that are

strictly divided into equally distributed

classes (Imdadullah, 2012). In table 3,

leaves width under 8-9 range occurred very

often with 57 repentances. On the other

hand, 12-13 range of width leaves only

tallied 5 inquiries.

Table 3. Frequency Distribution Table of Caesalpinia pulcherrima L.Width

Classes Frequency Midpoint Lower

Limit

Upper

Limit

Relative

Frequency

%

frequency

CF< CF> Rank

6-7 12 6.5 5.5 7.5 0.12 12 % 12 100 3

8-9 57 8.5 7.5 9.5 0.57 57 % 69 88 1

10-11 26 10.5 9.5 11.5 0.26 26 % 95 31 2

12-13 5 12.5 11.5 13.5 0.05 5 % 100 5 4

Table 4. Frequency Distribution Table of Caesalpinia pulcherrima L. Length

Classes Frequency Midpoint Lower

Limit

Upper

Limit

Relative

Frequency

%

frequency

CF< CF> Rank

11-12 8 11.5 10.5 12.5 0.08 8 % 8 100 5

13-14 18 13.5 12.5 14.5 0.18 18 % 26 92 3

15-16 39 15.5 14.5 16.5 0.39 39 % 65 74 1

17-18 19 17.5 16.5 18.5 0.19 19 % 84 35 2

19-20 14 19.5 18.5 20.5 0.14 14 % 98 16 4

21-22 2 21.5 20.5 22.5 0.02 2 % 100 2 6

0

5

10

15

20

25

0 5 10 15

Len

gth

Width

Frequency distribution table of

leaves length of C. pulcherrima L. was

shown in Table 4. Leaves length under the

range 15-16 befall for 39 times. The mean

and the frequent value range are therefore

relatively same as it depicts the expected

value that will occur in the data.

Linear Regression

Linear regression analysis is often

used to predict the (average) numerical

value of Y for a giver value of X using a

straight line, which is called the regression

line. The regression equation was

represented by the equation:

𝒚 = 𝒂 + 𝒃𝒙

Linear regression does not predict

the exact value of the variable but instead,

predicts the nearest possible value. It is also

used to understand which among the

independent variables are related to the

dependent variable, and to explore the forms

of these relationships (Yule, G. 1897)

𝑎 = ∑ 𝑦

𝑛− 𝑏

∑ 𝑥

𝑛

= 1592

100− 1.29

903

100

= 15.92 − 11.62

𝒂 = 𝟒. 𝟑𝟎 (estimated intercept)

𝑏 = ∑ 𝑥𝑦 − 𝐶𝑇𝑥𝑦

∑ 2𝑥 − 𝐶𝑇𝑥

= 14624 − 14375.76

8347 − 8154.09

= 248.24

192.91

𝒃 = 𝟏. 𝟐𝟗 (slope of the line)

The computed value for a, 4.30,

would generally be the estimated intercept

that would lie on the y-axis. And the slope,

b =1.29.

𝑟 = ∑ 𝑥𝑦 − 𝐶𝑇𝑥𝑦

√(∑ 2𝑥 − 𝐶𝑇𝑥) − (∑ 2 − 𝐶𝑇𝑦)𝑦

=14624 − 14375.76

√(8347 − 8154.09)(25902 − 25344.64)

= 248.24

√(192.91)(557.36)

= 248.24

√107520.3176

=248.24

327.9029088

𝒓 = 𝟎. 𝟕𝟓𝟕𝟎𝟓𝟑𝟑𝟔𝟑𝟓 (Strong correlation)

y = 4.30 + 1.29x

Figure 1. Scatter Diagram of Caesalpinia pulcherrima L. Leaf

Length and Width

Correlation measures the strength of

the linear association between two

quantitative variables (Wang, 2011). The

scatter diagram supports the underlying

statement of the correlation coefficient that

it the length and width are strongly

correlated. The diagram above showed an

uphill direction going right that directly

depicts a positive result or correlation; that

is if either X (width) or Y (length) changes,

the other seems to change in a predictably

affected way.

Leaf Length-Width Ratio

To determine the relationship of

length and width the computed values for

the intercept, 4.30, and the slope, 1.29 were

substituted in the regression equation

y = a + bx. The x value was assumed as 1 to

correlate that for every 1 mm of width there

will be a corresponding value for length.

𝒚 = 𝒂 + 𝒃𝒙

𝒚 = 𝟒. 𝟑𝟎 + 𝟏. 𝟐𝟗 (𝟏)

𝒚 = 𝟓. 𝟓𝟗

The equation above showed that for

every 1 mm of width 5.59 mm of length is

its corresponding measure approximately.

Conclusion

Statistical analysis helps in

transforming random information into

comprehensive data that can translate to

meaningful statements and conclusion. In

this study, the relationship of leaves length

and width were analyzed and with the use of

linear regression and correlation it was

concluded that the length is directly

proportional to the width and vice versa. It

was supported by the value computed for the

correlation coefficient and the Pearson’s r

interpreted that length and width are

therefore strongly related to each other.

References

Beerling, D. J. and Fry, J. C. 1990. A

Comparison of the Accuracy,

Variability and Speed of Five

Different Methods for Estimating

Leaf Area. Ann. Bot., 65: 483-488.

Blanco FF and Folegatti MV. 2005.

Estimation of leaf area for

greenhouse cucumber by linear

measurements under salinity and

grafting. – Sci. Agric.(Piracicaba)

62: 305-309.

Cho, Y. Y., Oh, S., Oh, M. M. and

Son, J. E. 2007. Estimation of

Individual Leaf Area, Fresh Weight,

and Dry Weight of Hydroponically

Grown Cucumbers (Cucumis sativus

L.) Using Leaf Length, Width, and

SPAD Value. Sci. Hort., 111: 330-

334.

De Jesus WC, Vale FXR, Coelho RR

and Costa LC. 2001. Comparison of

two methods for estimating leaf area

index on common bean. – Agron. J.

93: 989-991.

Demirsoy H, Demirsoy L, Uzun S

and Ersoy B. 2004. Nondestructive

leaf area estimation in peach. – Eur.

J. Hort. Sci. 69: 144-146

Imdadullah, Muhammad.

"Frequency Distribution". Retrieved

December 11, 2015 from

http://itfeature.com/statistics/frequen

cy-distribution-table. itfeature.com.

Loeve, M. (1977) "Probability

Theory", Graduate Texts in

Mathematics, Volume 45, 4th

edition, Springer-Verlag, p. 12.

Robbins, S. N. and Pharr, D. M.

1987. Leaf Area Prediction Models

250 Odels for Cucumber Linear

Measurements. Hort. Sci., 22: 1264-

1266.

Yule, G. Udny (1897). "On the

Theory of Correlation". Journal of

the Royal Statistical Society

(Blackwell Publishing) 60 (4): 812–

54. doi:10.2307/2979746. JSTOR

2979746.

Appendices

Table 1. Measurement of the Length and Width of Caesalpinia pulcherrima L.

Width (X) Length (Y) XY

1 10 17 170

2 6 12 72

3 10 16 160

4 8 15 120

5 11 19 209

6 9 16 144

7 8 15 120

8 9 15 135

9 8 14 112

10 11 16 176

11 9 19 171

12 7 11 77

13 9 16 144

14 7 11 77

15 10 17 170

16 7 13 91

17 8 11 88

18 9 11 99

19 9 11 99

20 9 13 117

21 10 17 170

22 9 15 135

23 13 20 260

24 11 19 209

25 10 18 180

26 11 20 220

27 9 15 135

28 8 16 128

29 10 19 190

30 9 17 153

31 9 18 162

32 10 18 180

33 9 11 99

34 9 16 144

35 12 20 240

36 9 18 162

37 13 21 273

38 9 16 144

39 10 19 190

40 9 17 153

41 9 17 153

42 10 17 170

43 10 19 190

44 10 20 200

45 7 14 98

46 8 16 128

47 9 17 153

48 10 18 180

49 8 14 112

50 9 17 153

51 9 16 144

52 11 18 209

53 10 17 170

54 8 16 128

55 9 16 144

56 11 16 176

57 11 19 209

58 11 20 220

59 9 18 162

60 10 18 180

61 12 18 216

62 8 15 120

63 7 14 98

64 9 16 144

65 9 16 144

66 8 15 120

67 8 14 112

68 9 16 144

69 8 15 120

70 8 15 120

71 10 19 190

72 9 16 144

73 8 15 120

74 8 14 112

75 10 16 160

76 9 18 162

77 10 16 160

78 9 16 144

79 10 16 160

80 8 14 112

81 8 11 88

82 9 13 117

83 9 16 144

84 8 16 128

85 7 14 98

86 9 14 126

87 9 16 144

88 9 16 144

89 7 14 98

90 13 21 273

91 7 14 98

92 7 14 98

93 8 15 120

94 7 14 98

95 8 15 120

96 8 16 128

97 8 16 128

98 7 13 91

99 8 13 104

100 8 15 120

Total 903 1592 14 624

Table 2. Descriptive Statistics Caesalpinia pulcherrima L. Width

i Xi X2

|Xi - �̅�| |Xi - �̅�|2

1 6 36 3.03 9.18 2 7 49 2.03 4.12 3 7 49 2.03 4.12 4 7 49 2.03 4.12 5 7 49 2.03 4.12 6 7 49 2.03 4.12 7 7 49 2.03 4.12 8 7 49 2.03 4.12 9 7 49 2.03 4.12

10 7 49 2.03 4.12 11 7 49 2.03 4.12 12 7 49 2.03 4.12 13 8 64 1.03 1.06 14 8 64 1.03 1.06 15 8 64 1.03 1.06 16 8 64 1.03 1.06 17 8 64 1.03 1.06 18 8 64 1.03 1.06 19 8 64 1.03 1.06 20 8 64 1.03 1.06 21 8 64 1.03 1.06 22 8 64 1.03 1.06 23 8 64 1.03 1.06 24 8 64 1.03 1.06 25 8 64 1.03 1.06 26 8 64 1.03 1.06 27 8 64 1.03 1.06 28 8 64 1.03 1.06 29 8 64 1.03 1.06 30 8 64 1.03 1.06 31 8 64 1.03 1.06 32 8 64 1.03 1.06 33 8 64 1.03 1.06 34 8 64 1.03 1.06 35 8 64 1.03 1.06 36 8 64 1.03 1.06 37 9 81 0.03 0.0009 38 9 81 0.03 0.0009

39 9 81 0.03 0.0009 40 9 81 0.03 0.0009 41 9 81 0.03 0.0009 42 9 81 0.03 0.0009 43 9 81 0.03 0.0009 44 9 81 0.03 0.0009 45 9 81 0.03 0.0009 46 9 81 0.03 0.0009 47 9 81 0.03 0.0009 48 9 81 0.03 0.0009 49 9 81 0.03 0.0009 50 9 81 0.03 0.0009 51 9 81 0.03 0.0009 52 9 81 0.03 0.0009 53 9 81 0.03 0.0009 54 9 81 0.03 0.0009 55 9 81 0.03 0.0009 56 9 81 0.03 0.0009 57 9 81 0.03 0.0009 58 9 81 0.03 0.0009 59 9 81 0.03 0.0009 60 9 81 0.03 0.0009 61 9 81 0.03 0.0009 62 9 81 0.03 0.0009 63 9 81 0.03 0.0009 64 9 81 0.03 0.0009 65 9 81 0.03 0.0009 66 9 81 0.03 0.0009 67 9 81 0.03 0.0009 68 9 81 0.03 0.0009 69 9 81 0.03 0.0009 70 10 100 0.97 0.94 71 10 100 0.97 0.94 72 10 100 0.97 0.94 73 10 100 0.97 0.94 74 10 100 0.97 0.94 75 10 100 0.97 0.94 76 10 100 0.97 0.94 77 10 100 0.97 0.94 78 10 100 0.97 0.94 79 10 100 0.97 0.94 80 10 100 0.97 0.94 81 10 100 0.97 0.94 82 10 100 0.97 0.94 83 10 100 0.97 0.94 84 10 100 0.97 0.94

85 10 100 0.97 0.94 86 10 100 0.97 0.94 87 10 100 0.97 0.94 88 11 121 1.97 3.88 89 11 121 1.97 3.88 90 11 121 1.97 3.88 91 11 121 1.97 3.88 92 11 121 1.97 3.88 93 11 121 1.97 3.88 94 11 121 1.97 3.88 95 11 121 1.97 3.88 96 12 144 2.97 8.82 97 12 144 2.97 8.82 98 13 169 3.97 15.76 99 13 169 3.97 15.76 100 13 169 3.97 15.76

Total 903 8347 102.14 192.91

Table 3. Descriptive Statistics of Caesalpinia pulcherrima L. Length

i Xi X2

|Xi - �̅�| |Xi - �̅�|2

1 11 121 4.92 24.21 2 11 121 4.92 24.21 3 11 121 4.92 24.21 4 11 121 4.92 24.21 5 11 121 4.92 24.21 6 11 121 4.92 24.21 7 11 121 4.92 24.21 8 12 144 3.92 15.37 9 13 169 2.92 8.53 10 13 169 2.92 8.53 11 13 169 2.92 8.53 12 13 169 2.92 8.53 13 13 169 2.92 8.53 14 14 196 1.92 3.69 15 14 196 1.92 3.69 16 14 196 1.92 3.69 17 14 196 1.92 3.69 18 14 196 1.92 3.69 19 14 196 1.92 3.69 20 14 196 1.92 3.69 21 14 196 1.92 3.69 22 14 196 1.92 3.69 23 14 196 1.92 3.69 24 14 196 1.92 3.69 25 14 196 1.92 3.69

26 14 196 1.92 3.69 27 15 225 0.92 0.85 28 15 225 0.92 0.85 29 15 225 0.92 0.85 30 15 225 0.92 0.85 31 15 225 0.92 0.85 32 15 225 0.92 0.85 33 15 225 0.92 0.85 34 15 225 0.92 0.85 35 15 225 0.92 0.85 36 15 225 0.92 0.85 37 15 225 0.92 0.85 38 15 225 0.92 0.85 39 15 225 0.92 0.85 40 16 256 0.08 0.0064 41 16 256 0.08 0.0064 42 16 256 0.08 0.0064 43 16 256 0.08 0.0064 44 16 256 0.08 0.0064 45 16 256 0.08 0.0064 46 16 256 0.08 0.0064 47 16 256 0.08 0.0064 48 16 256 0.08 0.0064 49 16 256 0.08 0.0064 50 16 256 0.08 0.0064 51 16 256 0.08 0.0064 52 16 256 0.08 0.0064 53 16 256 0.08 0.0064 54 16 256 0.08 0.0064 55 16 256 0.08 0.0064 56 16 256 0.08 0.0064 57 16 256 0.08 0.0064 58 16 256 0.08 0.0064 59 16 256 0.08 0.0064 60 16 256 0.08 0.0064 61 16 256 0.08 0.0064 62 16 256 0.08 0.0064 63 16 256 0.08 0.0064 64 16 256 0.08 0.0064 65 16 256 0.08 0.0064 66 17 289 1.08 1.17 67 17 289 1.08 1.17 68 17 289 1.08 1.17 69 17 289 1.08 1.17 70 17 289 1.08 1.17 71 17 289 1.08 1.17

72 17 289 1.08 1.17 73 17 289 1.08 1.17 74 17 289 1.08 1.17 75 17 289 1.08 1.17 76 18 324 2.08 4.33 77 18 324 2.08 4.33 78 18 324 2.08 4.33 79 18 324 2.08 4.33 80 18 324 2.08 4.33 81 18 324 2.08 4.33 82 18 324 2.08 4.33 83 18 324 2.08 4.33 84 18 324 2.08 4.33 85 19 361 3.08 9.49 86 19 361 3.08 9.49 87 19 361 3.08 9.49 88 19 361 3.08 9.49 89 19 361 3.08 9.49 90 19 361 3.08 9.49 91 19 361 3.08 9.49 92 19 361 3.08 9.49 93 19 361 3.08 9.49 94 20 400 4.08 16.65 95 20 400 4.08 16.65 96 20 400 4.08 16.65 97 20 400 4.08 16.65 98 20 400 4.08 16.65 99 21 441 5.08 25.81

100 21 441 5.08 25.81

Total 1592 25902 179.76 623.2556

Table 4. Stem and Leaf Diagram of Caesalpinia pulcherrima L. Width and Length

Width

0 6 77777 77777 7 88888 88888 88888 88888 8888 99999 99999 99999 99999 99999

99999 999

1 00000 00000 00000 000 11111 111 22 333

Length

1 11111 11 2 33333 44444 44444 444 55555 55555 555 66666 66666 66666 66666

66666 6 77777 77777 88888 8888 99999 9999

2 00000 11

Table 5. Regression Equation

Quantity Values

(1) n 100

(2) ∑ 𝑋 903

(3) ∑ 𝑌 1 592

(4) ∑ 𝑋2 8 347

(5) ∑ 𝑌2 25 902

(6) ∑ 𝑋𝑌 14 624

(7) (∑ 𝑋)2 815 409

(8) (∑ 𝑌)2 2 534 464

(9) (∑ 𝑋) ( ∑ 𝑌) 1 437 576

(10) 𝐶𝑇𝑥 8 154.09

(11) 𝐶𝑇𝑦 25 344.64

(12) 𝐶𝑇𝑥𝑦 14 375.76