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Correlation Examples in Statistics
In Statistics, the Correlation is used mainly to analyze the strength of the relationship between the variables that are under consideration and further it also measures if there is any relationship, one such common measures that are used in the field of statistics for correlation is the Pearson Correlation Coefficient. In correlation analysis, we estimate a sample correlation coefficient , more specifically the Pearson Product Moment correlation coefficient. The sample correlation coefficient, denoted r , ranges between -1 and +1 and quantifies the direction and strength of the linear association between the two variables. The correlation between two variables can be - positive (i.e., higher levels of one variable are associated with higher levels of the other) negative (i.e., higher levels of one variable are associated with lower levels of the other). The sign of the correlation coefficient indicates the direction of the association. The magnitude of the correlation coefficient indicates the strength of the association. Example # Vivek and Rupal are siblings, and Rupal is older than Vivek by three years. Sanjeev, their father, is a statistician, and he was interested in researching the linear relationship between height and weight. Hence, since their birth, he was noting their height and weight at various ages and arrived at the following:
Age
Rupal Vivek
Height (in foot) Weight (in Kgs) Height (in foot) Weight^ (in Kgs) 5 3.5 20 3.6 22 7 3.11 25 3.101 27 9 4.1 26 4.3 28 11 4.7 32 4.7 32 13 4.11 35 4.11 40 15 5.1 40 5.2 45 17 5.2 45 5.4 50 19 5.3 48 5.7 55 21 5.5 50 5.9 64 23 5.55 51 5.9 67 25 5.55 55 5.9 70 He tries to identify any correlation between age, height, and weight , and is there any differentiation between them? Solution: Firstly, plot a scatter chart, and the result for Rupal’s and Vivek’s age, height, and weight.
Temperature (in oC) Units Consumed Electricity Bill (in Rs) 24 80 2,490. 27 82 2,550. 30 84 2,610. 31 101 3,170. 34 110 3,890. 35 115 4,290. 38 140 6,390. 40 142 6,441. 42 156 7,155. 45 157 7,206. Solution: Let’s analyze this as well through a chart. Plotted electricity bills and temperature and noted their various points, A correlation between the temperature and electricity bill: when the temperature is cold , and the electricity bill is under control , which makes sense as the family would be using less of air condition and
when temperature increases , the use of air condition, geyser (natural hot spring) would increase which would hit them with a higher cost which is evident from the above graph where the electricity bill rises heavily. Thus, we can conclude that there is no linear relationship, but yes, there is a positive correlation. Hence, the family can again expect a bill amount for May month in the range of 6400 to 7000. Example - Correlation of Gestational Age and Birth Weight A small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams. Infant ID # Gestational Age (weeks) Birth Weight (grams) 1 34.7 1895 2 36.0 2030 3 29.3 1440 4 40.1 2835 5 35.7 3090 6 42.4 3827 7 40.3 3260 8 37.3 2690 9 40.9 3285 10 38.3 2920 11 38.5 3430 12 41.4 3657 13 39.7 3685 14 39.7 3345 15 41.1 3260 16 38.0 2680 17 38.7 2005 We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable. Each point represents an (x,y) pair (in this case the gestational age, measured in weeks, and the birth weight, measured in grams). Thus y=birth weight and x=gestational age. The data are displayed in a scatter diagram in the figure below.
University Expected = 180330/ Expected = Expected = 1140330/ Expected = Not live near University Observed = Expected = 180990/ Expected = Observed = Expected = 1140990/ Expected =
Sum 140 + 40 = 180 190 + 950 = 1140 1320 Degrees of freedom: DF = (r – 1) * (c – 1) Level of significance: .01 .05.
