{"id":37886,"date":"2026-01-19T12:59:20","date_gmt":"2026-01-19T11:59:20","guid":{"rendered":"https:\/\/www.hs-nordhausen.de\/?post_type=zusatzmaterial&p=37886"},"modified":"2026-01-20T10:18:33","modified_gmt":"2026-01-20T09:18:33","slug":"4-4-calculation-of-the-sampling-error-in-random-selection","status":"publish","type":"zusatzmaterial","link":"https:\/\/www.hs-nordhausen.de\/en\/zusatzmaterial\/4-4-berechnung-des-stichprobenfehlers-bei-der-zufallsauswahl\/","title":{"rendered":"4-4: Calculation of the sampling error in random selection"},"content":{"rendered":"

Marketing textbook, Keywords<\/span><\/p>\n\n\n\n

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Market research \u2192 Decision problems in the context of data collection \u2192 Selection procedure \u2192 Random selection (Chapter 4.2.3.1)<\/p>\n<\/div>\n\n\n

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In a partial survey, the characteristic of a feature that was surveyed in the sample is used to infer the corresponding characteristic of this feature in the population (e.g. proportion of consumers who use brand X).<\/strong><\/p>\n\n\n\n

However, such an \u201eextrapolation\u201c deviates to a greater or lesser extent from the \u201etrue\u201c expression of the characteristic of interest in the population. This \u201etrue\u201c value can be determined by means of a complete survey, but this is only possible or useful in exceptional cases. The so-called sampling error, which is also referred to as random error or maximum margin of error, indicates the random deviation of the expression of a characteristic in the sample from the \u201etrue\u201c expression of the characteristic in the population. The sampling error is calculated as follows:<\/p>\n\n\n\n

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Formula for calculating the sampling error<\/figcaption><\/figure>\n<\/div>\n\n\n
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\u00b1<\/mo>e<\/mi>=<\/mo>\u00b1<\/mo>t<\/mi>\u221a<\/mtext>(<\/mo>p<\/mi>\u00b7<\/mtext>q<\/mi>\/<\/mi>n<\/mi>)<\/mo><\/mrow>\u00b1e = \u00b1t \u221a(p - q \/ n)<\/annotation><\/semantics><\/math><\/div>\n<\/div><\/div>\n\n\n\n

e = sampling error, i.e. the fluctuation range (\u00b1) in per cent around the measured sample value within which the \u201etrue\u201c value (in the population) lies with a certain probability.<\/p>\n\n\n\n

t = indicator for the specified certainty of the inference from the sample to the population (the choice of t determines the significance level).<\/p>\n\n\n\n

The following table shows at which value of t the \u00bbtrue\u00ab value of the characteristic lies within the interval p \u00b1 e and with what probability.<\/p>\n\n\n\n

t-value<\/strong><\/th>Confidence probability (significance level) (in %)<\/strong><\/th><\/tr><\/thead>
1<\/strong><\/strong><\/td>68,3<\/td><\/tr>
1,96<\/strong><\/strong><\/td>95,0<\/td><\/tr>
2<\/strong><\/strong><\/td>95,5<\/td><\/tr>
3<\/strong><\/strong><\/td>99,7<\/td><\/tr>
3,29<\/strong><\/strong><\/td>99,9<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

p = proportion of people in the population who have a certain characteristic (e.g. proportion of users of brand X).<\/p>\n\n\n\n

q = proportion of people in the population who do not have a certain characteristic (e.g. proportion of non-users of brand X).
The product p - q - and thus also the sampling error - is greatest for a given sample size if p has a value of 50. This most unfavourable value in relation to the sampling error is always assumed if p is not known.<\/p>\n\n\n\n

n = sample size<\/p>\n\n\n\n

The following example illustrates the calculation of the sampling error: With a sample size of n = 400 and the assumption that p = 50 per cent, the sampling error is \u00b1 5 per cent if t = 2 is selected. Accordingly, the \u201etrue\u201c value of p, i.e. the proportion of users of brand X in the population, is between 45 and 55 per cent with a probability of 95.5 per cent (t = 2) (cf. Ter Hofte-Fankhauser\/W\u00e4lty, 2013, p. 50 f.) The calculated interval is also referred to as the confidence interval of the proportion value p. As the following figure shows, the size of the sampling error at a given significance level depends on the size of the sample on the one hand and on the value p on the other.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Sample size<\/strong>
n<\/strong><\/th>		P<\/strong>
q<\/strong><\/th>		50<\/strong>
50<\/strong><\/th>		45<\/strong>
55<\/strong><\/th>		40<\/strong>
60<\/strong><\/th>		35<\/strong>
65<\/strong><\/th>		30
70<\/strong><\/th>		25
75<\/strong><\/th>		20
80<\/strong><\/th>		...
...<\/th><\/tr><\/thead>
...<\/strong><\/td> <\/em><\/td>...<\/td>...<\/td>...<\/td>...<\/td>...<\/td>...<\/td>...<\/td>...<\/em><\/td><\/tr>
100<\/strong><\/em><\/strong><\/td> <\/em><\/td>10,0<\/em><\/td>9,9<\/em><\/td>9,8<\/em><\/td>9,5<\/em><\/td>9,2<\/em><\/td>8,7<\/em><\/td>8,0<\/em><\/td>...<\/em><\/td><\/tr>
200<\/strong><\/em><\/strong><\/td> <\/em><\/td>7,1<\/em><\/td>7,0<\/em><\/td>6,9<\/em><\/td>6,7<\/em><\/td>6,5<\/em><\/td>6,1<\/em><\/td>5,7<\/em><\/td>...<\/em><\/td><\/tr>
300<\/strong><\/em><\/strong><\/td> <\/em><\/td>5,8<\/em><\/td>5,7<\/em><\/td>5,7<\/em><\/td>5,5<\/em><\/td>5,3<\/em><\/td>5,0<\/em><\/td>4,6<\/em><\/td>...<\/em><\/td><\/tr>
400<\/strong><\/em><\/strong><\/td> <\/em><\/td>5,0<\/em><\/td>5,0<\/em><\/td>4,9<\/em><\/td>4,8<\/em><\/td>4,6<\/em><\/td>4,3<\/em><\/td>4,0<\/em><\/td>...<\/em><\/td><\/tr>
500<\/strong><\/em><\/strong><\/td> <\/em><\/td>4,5<\/em><\/td>4,4<\/em><\/td>4,4<\/em><\/td>4,3<\/em><\/td>4,1<\/em><\/td>3,9<\/em><\/td>3,6<\/em><\/td>...<\/em><\/td><\/tr>
600<\/strong><\/em><\/strong><\/td> <\/em><\/td>4,1<\/em><\/td>4,1<\/em><\/td>4,0<\/em><\/td>3,9<\/em><\/td>3,7<\/em><\/td>3,5<\/em><\/td>3,3<\/em><\/td>...<\/em><\/td><\/tr>
700<\/strong><\/em><\/strong><\/td> <\/em><\/td>3,8<\/em><\/td>3,8<\/em><\/td>3,7<\/em><\/td>3,6<\/em><\/td>3,5<\/em><\/td>3,3<\/em><\/td>3,0<\/em><\/td>...<\/em><\/td><\/tr>
800<\/strong><\/em><\/strong><\/td> <\/em><\/td>3,5<\/em><\/td>3,5<\/em><\/td>3,5<\/em><\/td>3,4<\/em><\/td>3,2<\/em><\/td>3,1<\/em><\/td>2,8<\/em><\/td>...<\/em><\/td><\/tr>
900<\/strong><\/em><\/strong><\/td> <\/em><\/td>3,3<\/em><\/td>3,3<\/em><\/td>3,3<\/em><\/td>3,2<\/em><\/td>3,1<\/em><\/td>2,9<\/em><\/td>2,7<\/em><\/td>...<\/em><\/td><\/tr>
1000<\/strong><\/em><\/strong><\/td> <\/em><\/td>3,2<\/em><\/td>3,1<\/em><\/td>3,1<\/em><\/td>3,0<\/em><\/td>2,9<\/em><\/td>2,7<\/em><\/td>2,5<\/em><\/td>...<\/em><\/td><\/tr>
...<\/strong><\/em><\/strong><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td>...<\/em><\/td><\/tr><\/tbody><\/table>
Sample error \u00b1e in per cent as a function of the sample size n and the distribution of p and q also in per cent (significance level: 95.5 per cent; t = 2).<\/figcaption><\/figure>\n\n\n\n

Strictly speaking, the calculation of the sampling error and the confidence interval is only permitted if the following condition is met: n - p - q \u2265 9 (Bortz\/Schuster 2010, p. 104 f.). This is the case in this example, as the following applies: 400 - 0.5 - 0.5 = 100. The sampling error can be calculated not only for percentage distributions, but also for sample averages of metric data (Homburg, 2015, p. 323 ff.).<\/p>\n<\/div><\/div>\n\n\n

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Sources:<\/strong><\/h4>\n\n\n\n
    \n
  • Bortz, J.\/Schuster, C.: Statistik f\u00fcr Human- und Sozialwissenschaftler, 7th edition, Berlin 2010.<\/li>\n\n\n\n
  • Homburg, C.: Marketing Management. Strategie - Instrumente - Umsetzung - Unternehmensf\u00fchrung, 7th edition, Wiesbaden 2020. <\/li>\n\n\n\n
  • Ter Hofte-Frankhauser, K.\/W\u00e4lty, H. F.: Marktforschung. Grundlagen mit zahlreichen Beispielen, Repetitionsfragen mit L\u00f6sungen und Glossar, 5th edition, Zurich 2013.<\/li>\n<\/ul>","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"template":"","meta":{"_piecal_is_recurring":false,"_piecal_recurring_interval":1,"_piecal_recurring_frequency":"","_piecal_recurring_exact_position":false,"_piecal_recurring_end":"","_piecal_color":"","_piecal_text_color":"","_piecal_global_color_master":false,"_piecal_rsets":"[]","_piecal_is_event":false,"_piecal_start_date":"","_piecal_end_date":"","_piecal_is_allday":false,"greyd_block_editor_preview":[],"pgc_sgb_lightbox_settings":""},"zusatzmaterial_category":[3289,3255],"zusatzmaterial_tag":[3290],"zielgruppen":[],"kontakt-zuordnung":[3203],"projekt":[1422],"institut-oder-einric":[683],"stg-zuordnung":[696],"zusatzbereiche":[1602],"fachbereich":[804],"wf_zusatzmaterial_folders":[3274],"ppma_author":[1400],"class_list":["post-37886","zusatzmaterial","type-zusatzmaterial","status-publish","hentry","zusatzmaterial_category-kapitel-4","zusatzmaterial_category-marketing-lehrbuch","zusatzmaterial_tag-kapitel-4","kontakt-zuordnung-stglassl","projekt-n-a","institut-oder-einric-sensoriklabor","stg-zuordnung-n-a","zusatzbereiche-_n-a","fachbereich-n-a"],"publishpress_future_workflow_manual_trigger":{"enabledWorkflows":[]},"_links":{"self":[{"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zusatzmaterial\/37886","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zusatzmaterial"}],"about":[{"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/types\/zusatzmaterial"}],"author":[{"embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/users\/1"}],"wp:attachment":[{"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/media?parent=37886"}],"wp:term":[{"taxonomy":"zusatzmaterial_category","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zusatzmaterial_category?post=37886"},{"taxonomy":"zusatzmaterial_tag","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zusatzmaterial_tag?post=37886"},{"taxonomy":"zielgruppen","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zielgruppen?post=37886"},{"taxonomy":"kontakt-zuordnung","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/kontakt-zuordnung?post=37886"},{"taxonomy":"projekt","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/projekt?post=37886"},{"taxonomy":"institut-oder-einric","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/institut-oder-einric?post=37886"},{"taxonomy":"stg-zuordnung","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/stg-zuordnung?post=37886"},{"taxonomy":"zusatzbereiche","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/zusatzbereiche?post=37886"},{"taxonomy":"fachbereich","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/fachbereich?post=37886"},{"taxonomy":"wf_zusatzmaterial_folders","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/wf_zusatzmaterial_folders?post=37886"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.hs-nordhausen.de\/en\/wp-json\/wp\/v2\/ppma_author?post=37886"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}