Presentation on theme: "Inferential Statistics 3: The Chi Square Test"— Presentation transcript:
1Inferential Statistics 3: The Chi Square Test Advanced Higher GeographyStatisticsInferential Statistics 3: The Chi Square Test
2Introduction (1)We often have occasions to make comparisons between two characteristics of something to see if they are linked or related to each other.One way to do this is to work out what we would expect to find if there was no relationship between them (the usual null hypothesis) and what we actually observe.
3Introduction (2)The test we use to measure the differences between what is observed and what is expected according to an assumed hypothesis is called the chi-square test.
4For Example Some null hypotheses may be: ‘there is no relationship between the height of the land and the vegetation cover’.‘there is no difference in the location of superstores and small grocers shops’‘there is no connection between the size of farm and the type of farm’
5ImportantThe chi square test can only be used on data that has the following characteristics:The frequency data mush have a precise numerical value and must be organised into categories or groups.The data must be in the form of frequenciesThe expected frequency in any one cell of the table must be greater than 5.The total number of observations must be greater than 20.
6Formula χ 2 = ∑ (O – E)2 E χ2 = The value of chi square O = The observed valueE = The expected value∑ (O – E)2 = all the values of (O – E) squared then added together
7Worked ExampleWrite down the NULL HYPOTHESIS and ALTERNATIVE HYPOTHESIS and set the LEVEL OF SIGNIFICANCE.NH ‘ there is no difference in the distribution of old established industries and food processing industries in the postal district of Leicester’AH ‘There is a difference in the distribution of old established industries and food processing industries in the postal district of Leicester’We will set the level of significance at 0.05.
8Table Time! Observed Frequencies (O) Construct a table with the information you have observed or obtained.Observed Frequencies (O)Post CodesLE1LE2LE3LE4LE5&LE6Row TotalOld Industry91310850Food Industry4352142Column Total1615192992(Note: that although there are 3 cells in the table that are not greater than 5, these are observed frequencies. It is only the expected frequencies that have to be greater than 5.)
9Now Work out the expected frequency. Expected frequency = row total x column totalGrand totalEg: expected frequency for old industry in LE1 = (50 x 13) / 92 = 7.07Post CodesLE1LE2LE3LE4LE5&LE6Row TotalOld Industry7.07Food IndustryColumn Total
10Check your answers Post Codes LE1 LE2 LE3 LE4 LE5&LE6 Row Total Old Industry7.078.708.1510.3315.7650Food Industry5.937.306.858.6713.2442Column Total131615192992
11Now (O – E)2 E For each of the cells calculate. Eg: Old industry in LE1 is (9 – 7.07)2 / 7.07 = 0.53Post CodesLE1LE2LE3LE4LE5&LE6Row TotalOld Industry0.53Food IndustryColumn Total
12Check your answers Then Post Codes LE1 LE2 LE3 LE4 LE5&LE6 Old Industry0.532.130.420.013.82Food Industry0.632.540.504.55ThenAdd up all of the above numbers to obtain the value for chi square: χ2 =
13FinallyLook up the significance tables. These will tell you whether to accept the null hypothesis or reject it.The number of degrees of freedom to use is: the number of rows in the table minus 1, multiplied by the number of columns minus 1. This is (2-1) x (5-1) = 1 x 4 = 4 degrees of freedom.We find that our answer of is greater than the critical value of 9.49 (for 4 degrees of freedom and a significance level of 0.05) and so we reject the null hypothesis.
14Now you have to look for geographical factors to explain your findings In other words‘The distribution of old established industry and food processing industries in Leicester is significantly different.’The hard bit!Now you have to look for geographical factors to explain your findings