Measuring normal distribution for weighted data

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I want to measure how normally distributed systolic blood pressures are for various large groups of people. There is a problem. Blood pressure values tend to end in 0. If the actual blood pressure is 134/92, the person records 130/90. Then, if they don't round down to 0, they tend to use even numbers instead of odd numbers. When a machine does the calculation, there is no weight added to zeros or even numbers. My frequency histogram has large spikes at the 10's, more spikes on all even numbers, and then lower counts for the odds. Therefore, if I use a standard fit measure (like Shapiro), it says it isn't normal at all. Is there a standard way to measure the normal fit for weighted data like this?

Typical data, from the request. This is just one of the groups, but it is very typical.

60  332
61  163
62  220
63  180
64  291
65  184
66  255
67  254
68  484
69  317
70  1335
71  439
72  1162
73  539
74  1062
75  711
76  1364
77  902
78  3004
79  1184
80  9963
81  1728
82  7370
83  2353
84  7493
85  3504
86  8073
87  4004
88  15385
89  4794
90  40668
91  7608
92  25324
93  9696
94  20924
95  12848
96  27513
97  14588
98  49466
99  18227
100 117119
101 25746
102 72973
103 27036
104 61717
105 31175
106 52934
107 32740
108 75508
109 35226
110 189954
111 37166
112 100084
113 39226
114 68546
115 46738
116 79198
117 44110
118 138691
119 46057
120 216560
121 45453
122 137759
123 45110
124 108539
125 46194
126 86105
127 42436
128 107570
129 40359
130 158431
131 40062
132 92663
133 39299
134 62746
135 41077
136 63258
137 34957
138 82864
139 30463
140 109501
141 25979
142 62505
143 26727
144 43459
145 26801
146 35002
147 21967
148 39155
149 20067
150 48416
151 17000
152 26781
153 15657
154 21041
155 14628
156 18240
157 12146
158 19963
159 10676
160 30784
161 8796
162 13938
163 7190
164 10724
165 7037
166 7848
167 5940
168 9222
169 4900
170 11461
171 4250
172 6174
173 4100
174 4590
175 3586
176 4391
177 3209
178 4761
179 2845
180 7708
181 2471
182 3321
183 2204
184 2821
185 2085
186 2386
187 1912
188 2319
189 1690
190 3037
191 1288
192 1610
193 1148
194 1264
195 1001
196 1076
197 965
198 1225
199 814
200 1594
201 566
202 686
203 480
204 659
205 421
206 460
207 374
208 444
209 323
210 781
211 326
212 314
213 277
214 308
215 260
216 274
217 185
218 310
219 205
220 499
221 279
222 192
223 139
224 213
225 144
226 160
227 127
228 162
229 121
230 223
231 103
232 106
233 101
234 66
235 119
236 95
237 59
238 175
239 48
240 153
241 73
242 71
243 50
244 47
245 57
246 36
247 61
248 60
249 29
250 77
251 39
252 58
253 30
254 30
255 46
256 27
257 40
258 48
259 19
260 59
261 20
262 36
263 18
264 30
265 27
266 15
267 21
268 34
269 28
270 23
271 13
272 37
273 12
274 18
275 24
276 7
277 18
278 15
279 16
280 21
281 2
282 6
283 3
284 12
285 1
286 4
287 2
288 4
289 4
290 4
291 6
292 7
293 5
294 6
295 4
296 3
297 3
298 11
299 6
300 6