Data in Brief ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 1 Contents lists available at ScienceDirect 2 3 4 Data in Brief 5 6 7 journal homepage: www.elsevier.com/locate/dib 8 9 Data Article 10 11 fi 12 Q1 Datasets on the statistical properties of the rst 13 3000 squared positive integers 14 15 Hilary I. Okagbue a,n, Muminu O. Adamu b, 16 a a 17 Pelumi E. Oguntunde , Abiodun A. Opanuga , c a 18 Ayodele A. Adebiyi , Sheila A. Bishop 19 Q5 a Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria 20 b Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria 21 c Department of Computer and Information Sciences, Covenant University, Canaanland, Ota, Nigeria 22 23 24 article info abstract 25 26 Article history: The data in this article are as a result of a quest to uncover alter- 27 Received 2 May 2017 native research routes of deepening researchers’ understanding of 28 Received in revised form integers apart from the traditional number theory approach. 12 September 2017 29 Hence, the article contains the statistical properties of the digits Accepted 26 September 2017 fi 30 sum of the rst 3000 squared positive integers. The data describes the various statistical tools applied to reveal different statistical 31 Keywords: and random nature of the digits sum of the first 3000 squared 32 Positive integer positive integers. Digits sum here implies the sum of all the digits Digits sum 33 that make up the individual integer. Harrell-Davis quantiles 34 & 2017 The Authors. Published by Elsevier Inc. This is an open Boxplots 35 Bootstrap access article under the CC BY license 36 M-estimators (http://creativecommons.org/licenses/by/4.0/). 37 Confidence intervals 38 Curve estimation fi 39 Model t 40 41 42 Specifications Table 43 44 45 Subject area Mathematics 46 More specific subject area Number Statistics, Computational number theory 47 Type of data Tables and Figures 48 49 50 n Corresponding author. 51 E-mail address: [email protected] (H.I. Okagbue). 52 http://dx.doi.org/10.1016/j.dib.2017.09.055 53 2352-3409/& 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license 54 (http://creativecommons.org/licenses/by/4.0/). Please cite this article as: H.I. Okagbue, et al., Datasets on the statistical properties of the first 3000 squared positive integers, Data in Brief (2017), http://dx.doi.org/10.1016/j.dib.2017.09.055i 2 H.I. Okagbue et al. / Data in Brief ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 55 How data was acquired The raw data is available in mathematical literature 56 Data format Analyzed 57 Experimental factors Zero and negative integers were not considered 58 Experimental features Exploratory data analysis, mathematical computation 59 Data source location Covenant University Mathematics Laboratory, Ota, Nigeria 60 Data accessibility All the data are in this data article 61 62 63 64 Value of the data 65 66 The data provides the exploratory statistics of digits sum of squared positive integers and their 67 subsets. 68 This technique of analysis can be used in data reduction. 69 The data analysis can be applied to other known numbers. 70 The data when completely analyzed can help deepen the understanding of the random nature of 71 integers. 72 73 74 75 1. Data 76 77 The data provides a description of the statistical properties of the digits sum of the first 3000 78 squared positive integers and the subsets. The subsets are the even and odd positive integers. The 79 subsets are equivalence and their descriptive statistics are summarized in Figs. 1–3: 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 Fig. 1. The summary statistics of the digits sum of squared positive integers. Remark: The gaps observed in the histogram are 108 because the digits sum of squared positive integers cannot yield some numbers such as: 2, 3, 5, 6, 8, 11, 12, 14, 15 and so on. Please cite this article as: H.I. Okagbue, et al., Datasets on the statistical properties of the first 3000 squared positive integers, Data in Brief (2017), http://dx.doi.org/10.1016/j.dib.2017.09.055i H.I. Okagbue et al. / Data in Brief ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 Fig. 2. The summary statistics of the digits sum of squared even positive integers. Remark: It can be seen that the mean and 136 median of the data set are almost the same. 137 138 139 2. Experimental design, materials and methods 140 141 Q2 The digits sum or digital sum of integers has been a subject of interest because of its application in 142 cryptography, primality testing, random number generation and data reduction. Details on the origin, 143 theories and applications of the digits sum of squared positive integers, integers and other important 144 number sequences can be found in [1–28]. Recently digits sum and digital root have been applied in 145 the analysis of lotto results [29]. 146 147 2.1. Exploratory data analysis 148 149 The true nature of the percentiles are shown using the Harrell-Davis quantile which is a better 150 estimator and a measure of variability because it makes use of the data in totality rather than the 151 percentiles that are based on order statistics. The Harrell-Davis quantile of the digits sum of square of 152 153 positive integers is shown in Fig. 4. fi 154 Bootstrap methods are useful in construction of highly accurate and reliable con dence intervals 155 (C.Is) for unknown and complicated probability distributions. The data for was resampled many times 156 and C.Is was generated for the mean and the standard deviation. Bootstrap results varied slightly with 157 the observed mean and standard deviation and convergence occurs as the confidence level increases. 158 These are shown in Tables 1 and 2: 159 The bootstrap estimate of the mean is closed to the observed one. However, the median remained 160 unchanged. This is an evidence of the robustness and the resistant nature of the median against 161 undue influence of outliers. This is also in agreement with the bootstrap confidence limits. The 162 summary is shown in Table 3. Please cite this article as: H.I. Okagbue, et al., Datasets on the statistical properties of the first 3000 squared positive integers, Data in Brief (2017), http://dx.doi.org/10.1016/j.dib.2017.09.055i 4 H.I. Okagbue et al. / Data in Brief ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 Fig. 3. The summary statistics of the digits sum of squared odd positive integers. Remark: Here, the mean and median of the 189 data set are the same. 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 Fig. 4. Harrell-Davis quantiles. 208 209 The M-Estimators are checked for the convergence to the mean or the median. The M-Estimators 210 are robust and resistant to the undue effect of outliers. Technically, an M-Estimator can be assumed as 211 the fixed point of the estimating function. The results of the M-estimator for the digits sum of the first 212 3000 squared positive integer is summarized in Table 4. 213 The boxplot is an exploratory data analysis tool used to display graphically, the quantiles of a given 214 numerical data. Outliers or extreme values are easily precipitated from the data and displayed graphically. 215 The boxplots of the digits sums of squared positive integers and their subsets are shown in Fig. 5: 216 Please cite this article as: H.I. Okagbue, et al., Datasets on the statistical properties of the first 3000 squared positive integers, Data in Brief (2017), http://dx.doi.org/10.1016/j.dib.2017.09.055i H.I. Okagbue et al. / Data in Brief ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5 217 Table 1 218 The bootstrap confidence interval for the mean of the digits sum of square of positive integers. 219 Confidence level (%) Lower limit Upper limit 220 221 99 27.02 27.76 222 98 27.03 27.77 223 97 27.07 27.75 96 27.08 27.72 224 95 27.10 27.70 225 94 27.12 27.68 226 93 27.12 27.70 227 92 27.12 27.66 228 91 27.12 27.66 90 27.14 27.64 229 230 231 Table 2 232 The bootstrap confidence interval for the standard deviation of the digits sum of square of 233 positive integers. 234 Confidence level (%) Lower limit Upper limit 235 236 99 8.22 8.763 237 98 8.246 8.735 238 97 8.262 8.715 96 8.261 8.709 239 95 8.292 8.700 240 94 8.308 8.693 241 93 8.29 8.689 242 92 8.325 8.681 91 8.316 8.66 243 90 8.311 8.674 244 245 246 Table 3 247 Estimation results of bootstrap of the mean and median of digits sum of squared positive integers. 248 Statistic P1 P5 Q1 Q2 (estimate) Q3 P95 P99 S.D. I.Q.R. 249 250 Mean 27.039 27.14 27.278 27.398 27.487 27.639 27.712 0.15221 0.20933 251 Median 27 27 27 27 27 27 27 0 0 252 ¼ ; ¼ ; ¼ ; ¼ ; ¼ ; 253 P1 first percentile P5 fifth percentile Q1 first quartile Q2 second quartile or the estimate Q3 third quartile P95 ¼ ninety five percentile; P99 ¼ ninety nine percentile; S:D: ¼ standard deviation; I:Q:R: ¼ the inter quartile range: 254 255 256 257 Table 4 The M-estimators for the first 3000 squared positive integers.