{
  "url": "https://openlegislation.uk/eu-regulation-2017-654/annex-VII/appendix-3",
  "json_url": "https://openlegislation.uk/eu-regulation-2017-654/annex-VII/appendix-3.json",
  "legislation": "eur/2017/654",
  "legislation_title": "Commission Delegated Regulation (EU) 2017/654 of 19 December 2016 supplementing Regulation (EU) 2016/1628 of the European Parliament and of the Council with regard to technical and general requirements relating to emission limits and type-approval for internal combustion engines for non-road mobile machinery",
  "legislation_url": "https://openlegislation.uk/eu-regulation-2017-654",
  "path": "annex/VII/appendix/3",
  "kind": "appendix",
  "number": "3",
  "heading": null,
  "text": "Statistics\n1.\nArithmetic mean\nThe arithmetic mean, y – , shall be calculated by means of equation (7-155):\ny – = ∑ i = 1 N y i N\n(7-155)\n2.\nStandard deviation\nThe standard deviation for a non-biased (e.g., N–1) sample, σ, shall be calculated by means of equation (7-156):\nσ y = ∑ i = 1 N y i − y – 2 N − 1\n(7-156)\n3.\nRoot mean square\nThe root mean square, rmsy , shall be calculated by means of equation (7-157):\nrms y = 1 N ∑ i = 1 N y 2 i\n(7-157)\n4.\nt-test\nIt shall be determined if the data passes a t-test by means of the following equations and Table 7.8:\n(a) For an unpaired t-test, the t statistic and its number of degrees of freedom, v, shall be calculated by means of equations (7-158) and (7-159):\nt = y – ref − y – σ 2 ref N ref + σ 2 y N\n(7-158)\nv = σ 2 ref N ref + σ 2 y N 2 σ 2 ref ∕ N ref 2 N ref − 1 + σ 2 y ∕ N 2 N − 1\n(7-159)\n(b) For a paired t-test, the t statistic and its number of degrees of freedom, v, shall be calculated by means of equation (7-160), noting that the ε i are the errors (e.g., differences) between each pair of y ref i and yi :\nt = ε – × N σ ε\nv= N – 1\n(7-160)\n(c) Table 7.8 shall be used to compare t to the t crit values tabulated versus the number of degrees of freedom. If t is less than t crit, then t passes the t-test.\nTable 7.8\nCritical t values versus number of degrees of freedom, v\nv\nConfidence\n90 %\n95 %\n1\n6,314\n12,706\n2\n2,920\n4,303\n3\n2,353\n3,182\n4\n2,132\n2,776\n5\n2,015\n2,571\n6\n1,943\n2,447\n7\n1,895\n2,365\n8\n1,860\n2,306\n9\n1,833\n2,262\n10\n1,812\n2,228\n11\n1,796\n2,201\n12\n1,782\n2,179\n13\n1,771\n2,160\n14\n1,761\n2,145\n15\n1,753\n2,131\n16\n1,746\n2,120\n18\n1,734\n2,101\n20\n1,725\n2,086\n22\n1,717\n2,074\n24\n1,711\n2,064\n26\n1,706\n2,056\n28\n1,701\n2,048\n30\n1,697\n2,042\n35\n1,690\n2,030\n40\n1,684\n2,021\n50\n1,676\n2,009\n70\n1,667\n1,994\n100\n1,660\n1,984\n1 000+\n1,645\n1,960\nLinear interpolation shall be used to establish values not shown here.\n5.\nF-test\nThe F statistic shall be calculated by means of equation (7-161):\nF y = σ 2 y σ 2 ref\n(7-161)\n(a) For a 90 % confidence F-test, Table 7.9 is used to compare F to the F crit90 values tabulated versus (N–1) and (N ref–1). If F is less than F crit90, then F passes the F-test at 90 % confidence;\n(b) For a 95 % confidence F-test, Table 7.10 is used to compare F to the F crit95 values tabulated versus (N–1) and (N ref–1). If F is less than F crit95, then F passes the F-test at 95 % confidence.\nTable 7-9\nCritical F values, F crit90 , versus N – 1 and N ref – 1 at 90 per cent confidence\nN – 1\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n12\n15\n20\n24\n30\n40\n60\n120\n1000+\nN ref – 1\n1\n39,86\n49,50\n53,59\n55,83\n57,24\n58,20\n58,90\n59,43\n59,85\n60,19\n60,70\n61,22\n61,74\n62,00\n62,26\n62,52\n62,79\n63,06\n63,32\n2\n8,526\n9,000\n9,162\n9,243\n9,293\n9,326\n9,349\n9,367\n9,381\n9,392\n9,408\n9,425\n9,441\n9,450\n9,458\n9,466\n9,475\n9,483\n9,491\n3\n5,538\n5,462\n5,391\n5,343\n5,309\n5,285\n5,266\n5,252\n5,240\n5,230\n5,216\n5,200\n5,184\n5,176\n5,168\n5,160\n5,151\n5,143\n5,134\n4\n4,545\n4,325\n4,191\n4,107\n4,051\n4,010\n3,979\n3,955\n3,936\n3,920\n3,896\n3,870\n3,844\n3,831\n3,817\n3,804\n3,790\n3,775\n3,761\n5\n4,060\n3,780\n3,619\n3,520\n3,453\n3,405\n3,368\n3,339\n3,316\n3,297\n3,268\n3,238\n3,207\n3,191\n3,174\n3,157\n3,140\n3,123\n3,105\n6\n3,776\n3,463\n3,289\n3,181\n3,108\n3,055\n3,014\n2,983\n2,958\n2,937\n2,905\n2,871\n2,836\n2,818\n2,800\n2,781\n2,762\n2,742\n2,722\n7\n3,589\n3,257\n3,074\n2,961\n2,883\n2,827\n2,785\n2,752\n2,725\n2,703\n2,668\n2,632\n2,595\n2,575\n2,555\n2,535\n2,514\n2,493\n2,471\n8\n3,458\n3,113\n2,924\n2,806\n2,726\n2,668\n2,624\n2,589\n2,561\n2,538\n2,502\n2,464\n2,425\n2,404\n2,383\n2,361\n2,339\n2,316\n2,293\n9\n3,360\n3,006\n2,813\n2,693\n2,611\n2,551\n2,505\n2,469\n2,440\n2,416\n2,379\n2,340\n2,298\n2,277\n2,255\n2,232\n2,208\n2,184\n2,159\n10\n3,285\n2,924\n2,728\n2,605\n2,522\n2,461\n2,414\n2,377\n2,347\n2,323\n2,284\n2,244\n2,201\n2,178\n2,155\n2,132\n2,107\n2,082\n2,055\n11\n3,225\n2,860\n2,660\n2,536\n2,451\n2,389\n2,342\n2,304\n2,274\n2,248\n2,209\n2,167\n2,123\n2,100\n2,076\n2,052\n2,026\n2,000\n1,972\n12\n3,177\n2,807\n2,606\n2,480\n2,394\n2,331\n2,283\n2,245\n2,214\n2,188\n2,147\n2,105\n2,060\n2,036\n2,011\n1,986\n1,960\n1,932\n1,904\n13\n3,136\n2,763\n2,560\n2,434\n2,347\n2,283\n2,234\n2,195\n2,164\n2,138\n2,097\n2,053\n2,007\n1,983\n1,958\n1,931\n1,904\n1,876\n1,846\n14\n3,102\n2,726\n2,522\n2,395\n2,307\n2,243\n2,193\n2,154\n2,122\n2,095\n2,054\n2,010\n1,962\n1,938\n1,912\n1,885\n1,857\n1,828\n1,797\n15\n3,073\n2,695\n2,490\n2,361\n2,273\n2,208\n2,158\n2,119\n2,086\n2,059\n2,017\n1,972\n1,924\n1,899\n1,873\n1,845\n1,817\n1,787\n1,755\n16\n3,048\n2,668\n2,462\n2,333\n2,244\n2,178\n2,128\n2,088\n2,055\n2,028\n1,985\n1,940\n1,891\n1,866\n1,839\n1,811\n1,782\n1,751\n1,718\n17\n3,026\n2,645\n2,437\n2,308\n2,218\n2,152\n2,102\n2,061\n2,028\n2,001\n1,958\n1,912\n1,862\n1,836\n1,809\n1,781\n1,751\n1,719\n1,686\n18\n3,007\n2,624\n2,416\n2,286\n2,196\n2,130\n2,079\n2,038\n2,005\n1,977\n1,933\n1,887\n1,837\n1,810\n1,783\n1,754\n1,723\n1,691\n1,657\n19\n2,990\n2,606\n2,397\n2,266\n2,176\n2,109\n2,058\n2,017\n1,984\n1,956\n1,912\n1,865\n1,814\n1,787\n1,759\n1,730\n1,699\n1,666\n1,631\n20\n2,975\n2,589\n2,380\n2,249\n2,158\n2,091\n2,040\n1,999\n1,965\n1,937\n1,892\n1,845\n1,794\n1,767\n1,738\n1,708\n1,677\n1,643\n1,607\n21\n2,961\n2,575\n2,365\n2,233\n2,142\n2,075\n2,023\n1,982\n1,948\n1,920\n1,875\n1,827\n1,776\n1,748\n1,719\n1,689\n1,657\n1,623\n1,586\n20\n2,949\n2,561\n2,351\n2,219\n2,128\n2,061\n2,008\n1,967\n1,933\n1,904\n1,859\n1,811\n1,759\n1,731\n1,702\n1,671\n1,639\n1,604\n1,567\n23\n2,937\n2,549\n2,339\n2,207\n2,115\n2,047\n1,995\n1,953\n1,919\n1,890\n1,845\n1,796\n1,744\n1,716\n1,686\n1,655\n1,622\n1,587\n1,549\n24\n2,927\n2,538\n2,327\n2,195\n2,103\n2,035\n1,983\n1,941\n1,906\n1,877\n1,832\n1,783\n1,730\n1,702\n1,672\n1,641\n1,607\n1,571\n1,533\n25\n2,918\n2,528\n2,317\n2,184\n2,092\n2,024\n1,971\n1,929\n1,895\n1,866\n1,820\n1,771\n1,718\n1,689\n1,659\n1,627\n1,593\n1,557\n1,518\n26\n2,909\n2,519\n2,307\n2,174\n2,082\n2,014\n1,961\n1,919\n1,884\n1,855\n1,809\n1,760\n1,706\n1,677\n1,647\n1,615\n1,581\n1,544\n1,504\n27\n2,901\n2,511\n2,299\n2,165\n2,073\n2,005\n1,952\n1,909\n1,874\n1,845\n1,799\n1,749\n1,695\n1,666\n1,636\n1,603\n1,569\n1,531\n1,491\n28\n2,894\n2,503\n2,291\n2,157\n2,064\n1,996\n1,943\n1,900\n1,865\n1,836\n1,790\n1,740\n1,685\n1,656\n1,625\n1,593\n1,558\n1,520\n1,478\n29\n2,887\n2,495\n2,283\n2,149\n2,057\n1,988\n1,935\n1,892\n1,857\n1,827\n1,781\n1,731\n1,676\n1,647\n1,616\n1,583\n1,547\n1,509\n1,467\n30\n2,881\n2,489\n2,276\n2,142\n2,049\n1,980\n1,927\n1,884\n1,849\n1,819\n1,773\n1,722\n1,667\n1,638\n1,606\n1,573\n1,538\n1,499\n1,456\n40\n2,835\n2,440\n2,226\n2,091\n1,997\n1,927\n1,873\n1,829\n1,793\n1,763\n1,715\n1,662\n1,605\n1,574\n1,541\n1,506\n1,467\n1,425\n1,377\n60\n2,791\n2,393\n2,177\n2,041\n1,946\n1,875\n1,819\n1,775\n1,738\n1,707\n1,657\n1,603\n1,543\n1,511\n1,476\n1,437\n1,395\n1,348\n1,291\n120\n2,748\n2,347\n2,130\n1,992\n1,896\n1,824\n1,767\n1,722\n1,684\n1,652\n1,601\n1,545\n1,482\n1,447\n1,409\n1,368\n1,320\n1,265\n1,193\n1000+\n2,706\n2,303\n2,084\n1,945\n1,847\n1,774\n1,717\n1,670\n1,632\n1,599\n1,546\n1,487\n1,421\n1,383\n1,342\n1,295\n1,240\n1,169\n1,000\nTable 7-10\nCritical F values, F crit95 , versus N – 1 and N ref – 1 at 95 per cent confidence\nN – 1\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n12\n15\n20\n24\n30\n40\n60\n120\n1000+\nN ref – 1\n1\n161,4\n199,5\n215,7\n224,5\n230,1\n233,9\n236,7\n238,8\n240,5\n241,8\n243,9\n245,9\n248,0\n249,0\n250,1\n251,1\n252,2\n253,2\n254,3\n2\n18,51\n19,00\n19,16\n19,24\n19,29\n19,33\n19,35\n19,37\n19,38\n19,39\n19,41\n19,42\n19,44\n19,45\n19,46\n19,47\n19,47\n19,48\n19,49\n3\n10,12\n9,552\n9,277\n9,117\n9,014\n8,941\n8,887\n8,845\n8,812\n8,786\n8,745\n8,703\n8,660\n8,639\n8,617\n8,594\n8,572\n8,549\n8,526\n4\n7,709\n6,944\n6,591\n6,388\n6,256\n6,163\n6,094\n6,041\n5,999\n5,964\n5,912\n5,858\n5,803\n5,774\n5,746\n5,717\n5,688\n5,658\n5,628\n5\n6,608\n5,786\n5,410\n5,192\n5,050\n4,950\n4,876\n4,818\n4,773\n4,735\n4,678\n4,619\n4,558\n4,527\n4,496\n4,464\n4,431\n4,399\n4,365\n6\n5,987\n5,143\n4,757\n4,534\n4,387\n4,284\n4,207\n4,147\n4,099\n4,060\n4,000\n3,938\n3,874\n3,842\n3,808\n3,774\n3,740\n3,705\n3,669\n7\n5,591\n4,737\n4,347\n4,120\n3,972\n3,866\n3,787\n3,726\n3,677\n3,637\n3,575\n3,511\n3,445\n3,411\n3,376\n3,340\n3,304\n3,267\n3,230\n8\n5,318\n4,459\n4,066\n3,838\n3,688\n3,581\n3,501\n3,438\n3,388\n3,347\n3,284\n3,218\n3,150\n3,115\n3,079\n3,043\n3,005\n2,967\n2,928\n9\n5,117\n4,257\n3,863\n3,633\n3,482\n3,374\n3,293\n3,230\n3,179\n3,137\n3,073\n3,006\n2,937\n2,901\n2,864\n2,826\n2,787\n2,748\n2,707\n10\n4,965\n4,103\n3,708\n3,478\n3,326\n3,217\n3,136\n3,072\n3,020\n2,978\n2,913\n2,845\n2,774\n2,737\n2,700\n2,661\n2,621\n2,580\n2,538\n11\n4,844\n3,982\n3,587\n3,357\n3,204\n3,095\n3,012\n2,948\n2,896\n2,854\n2,788\n2,719\n2,646\n2,609\n2,571\n2,531\n2,490\n2,448\n2,405\n12\n4,747\n3,885\n3,490\n3,259\n3,106\n2,996\n2,913\n2,849\n2,796\n2,753\n2,687\n2,617\n2,544\n2,506\n2,466\n2,426\n2,384\n2,341\n2,296\n13\n4,667\n3,806\n3,411\n3,179\n3,025\n2,915\n2,832\n2,767\n2,714\n2,671\n2,604\n2,533\n2,459\n2,420\n2,380\n2,339\n2,297\n2,252\n2,206\n14\n4,600\n3,739\n3,344\n3,112\n2,958\n2,848\n2,764\n2,699\n2,646\n2,602\n2,534\n2,463\n2,388\n2,349\n2,308\n2,266\n2,223\n2,178\n2,131\n15\n4,543\n3,682\n3,287\n3,056\n2,901\n2,791\n2,707\n2,641\n2,588\n2,544\n2,475\n2,403\n2,328\n2,288\n2,247\n2,204\n2,160\n2,114\n2,066\n16\n4,494\n3,634\n3,239\n3,007\n2,852\n2,741\n2,657\n2,591\n2,538\n2,494\n2,425\n2,352\n2,276\n2,235\n2,194\n2,151\n2,106\n2,059\n2,010\n17\n4,451\n3,592\n3,197\n2,965\n2,810\n2,699\n2,614\n2,548\n2,494\n2,450\n2,381\n2,308\n2,230\n2,190\n2,148\n2,104\n2,058\n2,011\n1,960\n18\n4,414\n3,555\n3,160\n2,928\n2,773\n2,661\n2,577\n2,510\n2,456\n2,412\n2,342\n2,269\n2,191\n2,150\n2,107\n2,063\n2,017\n1,968\n1,917\n19\n4,381\n3,522\n3,127\n2,895\n2,740\n2,628\n2,544\n2,477\n2,423\n2,378\n2,308\n2,234\n2,156\n2,114\n2,071\n2,026\n1,980\n1,930\n1,878\n20\n4,351\n3,493\n3,098\n2,866\n2,711\n2,599\n2,514\n2,447\n2,393\n2,348\n2,278\n2,203\n2,124\n2,083\n2,039\n1,994\n1,946\n1,896\n1,843\n21\n4,325\n3,467\n3,073\n2,840\n2,685\n2,573\n2,488\n2,421\n2,366\n2,321\n2,250\n2,176\n2,096\n2,054\n2,010\n1,965\n1,917\n1,866\n1,812\n22\n4,301\n3,443\n3,049\n2,817\n2,661\n2,549\n2,464\n2,397\n2,342\n2,297\n2,226\n2,151\n2,071\n2,028\n1,984\n1,938\n1,889\n1,838\n1,783\n23\n4,279\n3,422\n3,028\n2,796\n2,640\n2,528\n2,442\n2,375\n2,320\n2,275\n2,204\n2,128\n2,048\n2,005\n1,961\n1,914\n1,865\n1,813\n1,757\n24\n4,260\n3,403\n3,009\n2,776\n2,621\n2,508\n2,423\n2,355\n2,300\n2,255\n2,183\n2,108\n2,027\n1,984\n1,939\n1,892\n1,842\n1,790\n1,733\n25\n4,242\n3,385\n2,991\n2,759\n2,603\n2,490\n2,405\n2,337\n2,282\n2,237\n2,165\n2,089\n2,008\n1,964\n1,919\n1,872\n1,822\n1,768\n1,711\n26\n4,225\n3,369\n2,975\n2,743\n2,587\n2,474\n2,388\n2,321\n2,266\n2,220\n2,148\n2,072\n1,990\n1,946\n1,901\n1,853\n1,803\n1,749\n1,691\n27\n4,210\n3,354\n2,960\n2,728\n2,572\n2,459\n2,373\n2,305\n2,250\n2,204\n2,132\n2,056\n1,974\n1,930\n1,884\n1,836\n1,785\n1,731\n1,672\n28\n4,196\n3,340\n2,947\n2,714\n2,558\n2,445\n2,359\n2,291\n2,236\n2,190\n2,118\n2,041\n1,959\n1,915\n1,869\n1,820\n1,769\n1,714\n1,654\n29\n4,183\n3,328\n2,934\n2,701\n2,545\n2,432\n2,346\n2,278\n2,223\n2,177\n2,105\n2,028\n1,945\n1,901\n1,854\n1,806\n1,754\n1,698\n1,638\n30\n4,171\n3,316\n2,922\n2,690\n2,534\n2,421\n2,334\n2,266\n2,211\n2,165\n2,092\n2,015\n1,932\n1,887\n1,841\n1,792\n1,740\n1,684\n1,622\n40\n4,085\n3,232\n2,839\n2,606\n2,450\n2,336\n2,249\n2,180\n2,124\n2,077\n2,004\n1,925\n1,839\n1,793\n1,744\n1,693\n1,637\n1,577\n1,509\n60\n4,001\n3,150\n2,758\n2,525\n2,368\n2,254\n2,167\n2,097\n2,040\n1,993\n1,917\n1,836\n1,748\n1,700\n1,649\n1,594\n1,534\n1,467\n1,389\n120\n3,920\n3,072\n2,680\n2,447\n2,290\n2,175\n2,087\n2,016\n1,959\n1,911\n1,834\n1,751\n1,659\n1,608\n1,554\n1,495\n1,429\n1,352\n1,254\n1000+\n3,842\n2,996\n2,605\n2,372\n2,214\n2,099\n2,010\n1,938\n1,880\n1,831\n1,752\n1,666\n1,571\n1,517\n1,459\n1,394\n1,318\n1,221\n1,000\n6.\nSlope\nThe least-squares regression slope, a 1y, shall be calculated by means of equation (7-162):\na 1 y = ∑ i = 1 N y i − y – × y ref i − y – ref ∑ i = 1 N y ref i − y – ref 2\n(7-162)\n7.\nIntercept\nThe least-squares regression intercept, a 0y, shall be calculated by means of equation (7-163):\na 0 y = y – − a 1 y × y – ref\n(7-163)\n8.\nStandard estimate of error\nThe standard estimate of error, SEE, shall be calculated by means of equation (7-164):\nSEE y = ∑ i = 1 N y i − a 0 y − a 1 y × y ref i 2 N − 2\n(7-164)\n9.\nCoefficient of determination\nThe coefficient of determination, r 2, shall be calculated by means of equation (7-165):\nr 2 y = 1 − ∑ i = 1 N y i − a 0 y − a 1 y × y ref i 2 ∑ i = 1 N y i − y – 2\n(7-165)",
  "extent": "E+W+S+N.I.",
  "status": null,
  "version": "current",
  "annotations": [
    {
      "type": "F",
      "label": "F1",
      "text": "Inserted by Commission Delegated Regulation (EU) 2018/989 of 18 May 2018 amending and correcting Delegated Regulation (EU) 2017/654 supplementing Regulation (EU) 2016/1628 of the European Parliament and of the Council with regard to technical and general requirements relating to emission limits and type-approval for internal combustion engines for non-road mobile machinery (Text with EEA relevance)."
    }
  ],
  "outstanding_effects": [],
  "provenance": {
    "source": "legislation.gov.uk",
    "source_url": "https://www.legislation.gov.uk/eur/2017/654/annex/VII/appendix/3",
    "retrieved_at": "2026-11-01T14:45:01.864924+00:00"
  },
  "licence": "Open Government Licence v3.0",
  "licence_url": "https://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/",
  "attribution": "Contains public sector information licensed under the Open Government Licence v3.0. Source: legislation.gov.uk, The National Archives."
}