how do you solve ANOVA for one way given the following data
Year Winner Length Seconds Speed Direction
1867 Ruthless 1.625 185 31.621622 Clockwise
1868 General Duke 1.625 182 32.142857 Clockwise
1869 Fenian 1.625 184.3 31.741725 Clockwise
1870 Kingfisher 1.625 176 33.238636 Clockwise
1871 Harry Bassett 1.625 179.5 32.590529 Clockwise
1872 Joe Daniels 1.625 178.2 32.828283 Clockwise
1873 Springbok 1.625 181.7 32.195927 Clockwise
1874 Saxon 1.5 159.5 33.855799 Clockwise
1875 Calvin 1.5 162.3 33.271719 Clockwise
1876 Algerine 1.5 160.5 33.64486 Clockwise
1877 Cloverbrook 1.5 166 32.53012 Clockwise
1878 Duke of Magenta 1.5 163.5 33.027523 Clockwise
1879 Spendthrift 1.5 162.7 33.18992 Clockwise
1880 Grenada 1.5 167 32.335329 Clockwise
1881 Saunterer 1.5 167 32.335329 Clockwise
1882 Forester 1.5 163 33.128834 Clockwise
1883 George Kinney 1.5 162.5 33.230769 Clockwise
1884 Panique 1.5 162 33.333333 Clockwise
1885 Tyrant 1.5 163 33.128834 Clockwise
1886 Inspector B 1.5 161 33.540373 Clockwise
1887 Hanover 1.5 163.5 33.027523 Clockwise
1888 Sir Dixon 1.5 160.3 33.686837 Clockwise
1889 Eric 1.5 167.3 32.277346 Clockwise
1890 Burlington 1.25 127.8 35.211268 Clockwise
1891 Foxford 1.25 128.8 34.937888 Clockwise
1892 Patron 1.25 137 32.846715 Clockwise
1893 Comanche 1.125 113.2 35.777385 Clockwise
1894 Henry of Navarre 1.125 116.5 34.763948 Clockwise
1895 Belmar 1.25 131.5 34.220532 Clockwise
1896 Hastings 1.375 144.5 34.256055 Clockwise
1897 Scottish Chieftain 1.375 143.3 34.542917 Clockwise
1898 Bowling Brook 1.375 152 32.565789 Clockwise
1899 Jean Bereaud 1.375 143 34.615385 Clockwise
1900 Ildrim 1.375 141.2 35.056657 Clockwise
1901 Commando 1.375 141 35.106383 Clockwise
1902 Masterman 1.375 142.6 34.712482 Clockwise
1903 Africander 1.375 141.7 34.932957 Clockwise
1904 Delhi 1.25 126.6 35.545024 Clockwise
1905 Tanya 1.25 128 35.15625 Clockwise
1906 Burgomaster 1.375 140 35.357143 Clockwise
1907 Peter Pan 1.375 NA NA Clockwise
1908 Colin 1.375 NA NA Clockwise
1909 Joe Madden 1.375 141.6 34.957627 Clockwise
1910 Sweep 1.375 142 34.859155 Clockwise
1913 Prince Eugene 1.375 138 35.869565 Clockwise
1914 Luke McLuke 1.375 140 35.357143 Clockwise
1915 The Finn 1.375 138.4 35.765896 Clockwise
1916 Friar Rock 1.375 142 34.859155 Clockwise
1917 Hourless 1.375 137.8 35.921626 Clockwise
1918 Johren 1.375 140.4 35.25641 Clockwise
1919 Sir Barton 1.375 137.4 36.026201 Clockwise
1920 Man o' War 1.375 134.2 36.885246 Clockwise
1921 Grey Lag 1.375 136.8 36.184211 counter-clockwise
1922 Pillory 1.375 138.8 35.662824 counter-clockwise
1923 Zev 1.375 139 35.611511 counter-clockwise
1924 Mad Play 1.375 138.8 35.662824 counter-clockwise
1925 American Flag 1.375 136.8 36.184211 counter-clockwise
1926 Crusader 1.5 152.2 35.479632 counter-clockwise
1927 Chance Shot 1.5 152.4 35.433071 counter-clockwise
1928 Vito 1.5 153.2 35.248042 counter-clockwise
1929 Blue Larkspur 1.5 152.8 35.340314 counter-clockwise
1930 Gallant Fox 1.5 151.6 35.620053 counter-clockwise
1931 Twenty Grand 1.5 149.6 36.096257 counter-clockwise
1932 Faireno 1.5 152.8 35.340314 counter-clockwise
1933 Hurryoff 1.5 152.6 35.386632 counter-clockwise
1934 Peace Chance 1.5 149.2 36.193029 counter-clockwise
1935 Omaha 1.5 150.6 35.856574 counter-clockwise
1936 Granville 1.5 150 36 counter-clockwise
1937 War Admiral 1.5 148.6 36.339166 counter-clockwise
1938 Pasteurized 1.5 149.4 36.144578 counter-clockwise
1939 Johnstown 1.5 149.6 36.096257 counter-clockwise
1940 Bimelech 1.5 149.6 36.096257 counter-clockwise
1941 Whirlaway 1.5 151 35.761589 counter-clockwise
1942 Shut Out 1.5 149.2 36.193029 counter-clockwise
1943 Count Fleet 1.5 148.2 36.437247 counter-clockwise
1944 Bounding Home 1.5 152.2 35.479632 counter-clockwise
1945 Pavot 1.5 150.2 35.952064 counter-clockwise
1946 Assault 1.5 150.8 35.809019 counter-clockwise
1947 Phalanx 1.5 149.4 36.144578 counter-clockwise
1948 Citation 1.5 148.2 36.437247 counter-clockwise
1949 Capot 1.5 150.2 35.952064 counter-clockwise
1950 Middleground 1.5 148.6 36.339166 counter-clockwise
1951 Counterpoint 1.5 149 36.241611 counter-clockwise
1952 One Count 1.5 150.2 35.952064 counter-clockwise
1953 Native Dancer 1.5 148.6 36.339166 counter-clockwise
1954 High Gun 1.5 150.8 35.809019 counter-clockwise
1955 Nashua 1.5 149 36.241611 counter-clockwise
1956 Needles 1.5 149.8 36.048064 counter-clockwise
1957 Gallant Man 1.5 146.6 36.834925 counter-clockwise
1958 Cavan 1.5 150.2 35.952064 counter-clockwise
1959 Sword Dancer 1.5 148.4 36.38814 counter-clockwise
1960 Celtic Ash 1.5 149.2 36.193029 counter-clockwise
1961 Sherluck 1.5 149.2 36.193029 counter-clockwise
1962 Jaipur 1.5 148.8 36.290323 counter-clockwise
1963 Chateaugay 1.5 150.2 35.952064 counter-clockwise
1964 Quadrangle 1.5 148.4 36.38814 counter-clockwise
1965 Hail To All 1.5 148.4 36.38814 counter-clockwise
1966 Amberoid 1.5 149.6 36.096257 counter-clockwise
1967 Damascus 1.5 148.8 36.290323 counter-clockwise
1968 Stage Door Johnny 1.5 147.2 36.684783 counter-clockwise
1969 Arts and Letters 1.5 148.8 36.290323 counter-clockwise
1970 High Echelon 1.5 154 35.064935 counter-clockwise
1971 Pass Catcher 1.5 150.4 35.904255 counter-clockwise
1972 Riva Ridge 1.5 148 36.486486 counter-clockwise
1973 Secretariat 1.5 144 37.5 counter-clockwise
1974 Little Current 1.5 149.2 36.193029 counter-clockwise
1975 Avatar 1.5 148.2 36.437247 counter-clockwise
1976 Bold Forbes 1.5 149 36.241611 counter-clockwise
1977 Seattle Slew 1.5 149.6 36.096257 counter-clockwise
1978 Affirmed 1.5 146.8 36.784741 counter-clockwise
1979 Coastal 1.5 148.6 36.339166 counter-clockwise
1980 Temperence Hill 1.5 149.8 36.048064 counter-clockwise
1981 Summing 1.5 149 36.241611 counter-clockwise
1982 Conquistador Cielo 1.5 148.2 36.437247 counter-clockwise
1983 Caveat 1.5 147.8 36.535859 counter-clockwise
1984 Swale 1.5 147.2 36.684783 counter-clockwise
1985 Creme Fraiche 1.5 147 36.734694 counter-clockwise
1986 Danzig Connection 1.5 149.8 36.048064 counter-clockwise
1987 Bet Twice 1.5 148.2 36.437247 counter-clockwise
1988 Risen Star 1.5 146.4 36.885246 counter-clockwise
1989 Easy Goer 1.5 146 36.986301 counter-clockwise
1990 Go And Go 1.5 147.2 36.684783 counter-clockwise
1991 Hansel 1.5 148.1 36.46185 counter-clockwise
1992 A.P. Indy 1.5 146.1 36.960986 counter-clockwise
1993 Colonial Affair 1.5 150 36 counter-clockwise
1994 Tabasco Cat 1.5 146.8 36.784741 counter-clockwise
1995 Thunder Gulch 1.5 152 35.526316 counter-clockwise
1996 Editor's Note 1.5 149 36.241611 counter-clockwise
1997 Touch Gold 1.5 148.8 36.290323 counter-clockwise
1998 Victory Gallop 1.5 149.2 36.193029 counter-clockwise
1999 Lemon Drop Kid 1.5 147.9 36.511156 counter-clockwise
2000 Commendable 1.5 151.2 35.714286 counter-clockwise
2001 Point Given 1.5 146.6 36.834925 counter-clockwise
2002 Sarava 1.5 149.7 36.072144 counter-clockwise
2003 Empire Maker 1.5 148.3 36.412677 counter-clockwise
2004 Birdstone 1.5 147.5 36.610169 counter-clockwise
2005 Afleet Alex 1.5 148.7 36.314728 counter-clockwise
2006 Jazil 1.5 147.9 36.511156 counter-clockwise
2007 Rags to Riches 1.5 148.7 36.314728 counter-clockwise
2008 Da'Tara 1.5 149.7 36.072144 counter-clockwise
2009 Summer Bird 1.5 147.5 36.610169 counter-clockwise
2010 Drosselmeyer 1.5 151.6 35.620053 counter-clockwise
2011 Ruler on Ice 1.5 150.9 35.785288 counter-clockwise
2012 Union Rags 1.5 150.4 35.904255 counter-clockwise
2013 Palace Malice 1.5 150.7 35.83278 counter-clockwise
2014 Tonalist 1.5 148.5 36.363636 counter-clockwise
2015 American Pharoah 1.5 146.7 36.809816 counter-clockwise
To solve the one-way ANOVA for the given data, follow these steps:
1. Define the null hypothesis (H0) and the alternative hypothesis (Ha).
- Null Hypothesis (H0): The mean speeds of horses in the clockwise and counter-clockwise directions are equal.
- Alternative Hypothesis (Ha): The mean speeds of horses in the clockwise and counter-clockwise directions are not equal.
2. Calculate the sum of squares total (SST) by summing the squared differences between each speed observation and the overall mean speed.
3. Calculate the sum of squares between (SSB) by summing the squared differences between the mean speeds of the two groups (clockwise and counter-clockwise) and the overall mean speed.
4. Calculate the sum of squares within (SSW) by summing the squared differences between each speed observation and the mean speed of its own group.
5. Calculate the degrees of freedom for the between-group variation (dfB) by subtracting 1 from the number of groups (2 in this case).
6. Calculate the degrees of freedom for the within-group variation (dfW) by subtracting the number of groups from the total number of observations.
7. Calculate the F-statistic by dividing the mean square between (MSB = SSB/dfB) by the mean square within (MSW = SSW/dfW).
8. Determine the critical F-value for your chosen level of significance (α) and degrees of freedom (dfB and dfW).
9. Compare the calculated F-statistic to the critical F-value to make a decision.
- If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis. There is evidence of a significant difference in mean speeds between the clockwise and counter-clockwise directions.
- If the calculated F-statistic is less than or equal to the critical F-value, fail to reject the null hypothesis. There is not enough evidence to conclude a significant difference in mean speeds between the clockwise and counter-clockwise directions.
Note: It is important to ensure that the data meets the assumptions of the one-way ANOVA, including independence of observations, normality, and homogeneous variances.
solve paired sample T test using the following data(you can just choice some of the data you want to use).
Year Winner Length Seconds Speed Direction
1867 Ruthless 1.625 185 31.621622 Clockwise
1868 General Duke 1.625 182 32.142857 Clockwise
1869 Fenian 1.625 184.3 31.741725 Clockwise
1870 Kingfisher 1.625 176 33.238636 Clockwise
1871 Harry Bassett 1.625 179.5 32.590529 Clockwise
1872 Joe Daniels 1.625 178.2 32.828283 Clockwise
1873 Springbok 1.625 181.7 32.195927 Clockwise
1874 Saxon 1.5 159.5 33.855799 Clockwise
1875 Calvin 1.5 162.3 33.271719 Clockwise
1876 Algerine 1.5 160.5 33.64486 Clockwise
1877 Cloverbrook 1.5 166 32.53012 Clockwise
1878 Duke of Magenta 1.5 163.5 33.027523 Clockwise
1879 Spendthrift 1.5 162.7 33.18992 Clockwise
1880 Grenada 1.5 167 32.335329 Clockwise
1881 Saunterer 1.5 167 32.335329 Clockwise
1882 Forester 1.5 163 33.128834 Clockwise
1883 George Kinney 1.5 162.5 33.230769 Clockwise
1884 Panique 1.5 162 33.333333 Clockwise
1885 Tyrant 1.5 163 33.128834 Clockwise
1886 Inspector B 1.5 161 33.540373 Clockwise
1887 Hanover 1.5 163.5 33.027523 Clockwise
1888 Sir Dixon 1.5 160.3 33.686837 Clockwise
1889 Eric 1.5 167.3 32.277346 Clockwise
1890 Burlington 1.25 127.8 35.211268 Clockwise
1891 Foxford 1.25 128.8 34.937888 Clockwise
1892 Patron 1.25 137 32.846715 Clockwise
1893 Comanche 1.125 113.2 35.777385 Clockwise
1894 Henry of Navarre 1.125 116.5 34.763948 Clockwise
1895 Belmar 1.25 131.5 34.220532 Clockwise
1896 Hastings 1.375 144.5 34.256055 Clockwise
1897 Scottish Chieftain 1.375 143.3 34.542917 Clockwise
1898 Bowling Brook 1.375 152 32.565789 Clockwise
1899 Jean Bereaud 1.375 143 34.615385 Clockwise
1900 Ildrim 1.375 141.2 35.056657 Clockwise
1901 Commando 1.375 141 35.106383 Clockwise
1902 Masterman 1.375 142.6 34.712482 Clockwise
1903 Africander 1.375 141.7 34.932957 Clockwise
1904 Delhi 1.25 126.6 35.545024 Clockwise
1905 Tanya 1.25 128 35.15625 Clockwise
1906 Burgomaster 1.375 140 35.357143 Clockwise
1907 Peter Pan 1.375 NA NA Clockwise
1908 Colin 1.375 NA NA Clockwise
1909 Joe Madden 1.375 141.6 34.957627 Clockwise
1910 Sweep 1.375 142 34.859155 Clockwise
1913 Prince Eugene 1.375 138 35.869565 Clockwise
1914 Luke McLuke 1.375 140 35.357143 Clockwise
1915 The Finn 1.375 138.4 35.765896 Clockwise
1916 Friar Rock 1.375 142 34.859155 Clockwise
1917 Hourless 1.375 137.8 35.921626 Clockwise
1918 Johren 1.375 140.4 35.25641 Clockwise
1919 Sir Barton 1.375 137.4 36.026201 Clockwise
1920 Man o' War 1.375 134.2 36.885246 Clockwise
1921 Grey Lag 1.375 136.8 36.184211 counter-clockwise
1922 Pillory 1.375 138.8 35.662824 counter-clockwise
1923 Zev 1.375 139 35.611511 counter-clockwise
1924 Mad Play 1.375 138.8 35.662824 counter-clockwise
1925 American Flag 1.375 136.8 36.184211 counter-clockwise
1926 Crusader 1.5 152.2 35.479632 counter-clockwise
1927 Chance Shot 1.5 152.4 35.433071 counter-clockwise
1928 Vito 1.5 153.2 35.248042 counter-clockwise
1929 Blue Larkspur 1.5 152.8 35.340314 counter-clockwise
1930 Gallant Fox 1.5 151.6 35.620053 counter-clockwise
1931 Twenty Grand 1.5 149.6 36.096257 counter-clockwise
1932 Faireno 1.5 152.8 35.340314 counter-clockwise
1933 Hurryoff 1.5 152.6 35.386632 counter-clockwise
1934 Peace Chance 1.5 149.2 36.193029 counter-clockwise
1935 Omaha 1.5 150.6 35.856574 counter-clockwise
1936 Granville 1.5 150 36 counter-clockwise
1937 War Admiral 1.5 148.6 36.339166 counter-clockwise
1938 Pasteurized 1.5 149.4 36.144578 counter-clockwise
1939 Johnstown 1.5 149.6 36.096257 counter-clockwise
1940 Bimelech 1.5 149.6 36.096257 counter-clockwise
1941 Whirlaway 1.5 151 35.761589 counter-clockwise
1942 Shut Out 1.5 149.2 36.193029 counter-clockwise
1943 Count Fleet 1.5 148.2 36.437247 counter-clockwise
1944 Bounding Home 1.5 152.2 35.479632 counter-clockwise
1945 Pavot 1.5 150.2 35.952064 counter-clockwise
1946 Assault 1.5 150.8 35.809019 counter-clockwise
1947 Phalanx 1.5 149.4 36.144578 counter-clockwise
1948 Citation 1.5 148.2 36.437247 counter-clockwise
1949 Capot 1.5 150.2 35.952064 counter-clockwise
1950 Middleground 1.5 148.6 36.339166 counter-clockwise
1951 Counterpoint 1.5 149 36.241611 counter-clockwise
1952 One Count 1.5 150.2 35.952064 counter-clockwise
1953 Native Dancer 1.5 148.6 36.339166 counter-clockwise
1954 High Gun 1.5 150.8 35.809019 counter-clockwise
1955 Nashua 1.5 149 36.241611 counter-clockwise
1956 Needles 1.5 149.8 36.048064 counter-clockwise
1957 Gallant Man 1.5 146.6 36.834925 counter-clockwise
1958 Cavan 1.5 150.2 35.952064 counter-clockwise
1959 Sword Dancer 1.5 148.4 36.38814 counter-clockwise
1960 Celtic Ash 1.5 149.2 36.193029 counter-clockwise
1961 Sherluck 1.5 149.2 36.193029 counter-clockwise
1962 Jaipur 1.5 148.8 36.290323 counter-clockwise
1963 Chateaugay 1.5 150.2 35.952064 counter-clockwise
1964 Quadrangle 1.5 148.4 36.38814 counter-clockwise
1965 Hail To All 1.5 148.4 36.38814 counter-clockwise
1966 Amberoid 1.5 149.6 36.096257 counter-clockwise
1967 Damascus 1.5 148.8 36.290323 counter-clockwise
1968 Stage Door Johnny 1.5 147.2 36.684783 counter-clockwise
1969 Arts and Letters 1.5 148.8 36.290323 counter-clockwise
1970 High Echelon 1.5 154 35.064935 counter-clockwise
1971 Pass Catcher 1.5 150.4 35.904255 counter-clockwise
1972 Riva Ridge 1.5 148 36.486486 counter-clockwise
1973 Secretariat 1.5 144 37.5 counter-clockwise
1974 Little Current 1.5 149.2 36.193029 counter-clockwise
1975 Avatar 1.5 148.2 36.437247 counter-clockwise
1976 Bold Forbes 1.5 149 36.241611 counter-clockwise
1977 Seattle Slew 1.5 149.6 36.096257 counter-clockwise
1978 Affirmed 1.5 146.8 36.784741 counter-clockwise
1979 Coastal 1.5 148.6 36.339166 counter-clockwise
1980 Temperence Hill 1.5 149.8 36.048064 counter-clockwise
1981 Summing 1.5 149 36.241611 counter-clockwise
1982 Conquistador Cielo 1.5 148.2 36.437247 counter-clockwise
1983 Caveat 1.5 147.8 36.535859 counter-clockwise
1984 Swale 1.5 147.2 36.684783 counter-clockwise
1985 Creme Fraiche 1.5 147 36.734694 counter-clockwise
1986 Danzig Connection 1.5 149.8 36.048064 counter-clockwise
1987 Bet Twice 1.5 148.2 36.437247 counter-clockwise
1988 Risen Star 1.5 146.4 36.885246 counter-clockwise
1989 Easy Goer 1.5 146 36.986301 counter-clockwise
1990 Go And Go 1.5 147.2 36.684783 counter-clockwise
1991 Hansel 1.5 148.1 36.46185 counter-clockwise
1992 A.P. Indy 1.5 146.1 36.960986 counter-clockwise
1993 Colonial Affair 1.5 150 36 counter-clockwise
1994 Tabasco Cat 1.5 146.8 36.784741 counter-clockwise
1995 Thunder Gulch 1.5 152 35.526316 counter-clockwise
1996 Editor's Note 1.5 149 36.241611 counter-clockwise
1997 Touch Gold 1.5 148.8 36.290323 counter-clockwise
1998 Victory Gallop 1.5 149.2 36.193029 counter-clockwise
1999 Lemon Drop Kid 1.5 147.9 36.511156 counter-clockwise
2000 Commendable 1.5 151.2 35.714286 counter-clockwise
2001 Point Given 1.5 146.6 36.834925 counter-clockwise
2002 Sarava 1.5 149.7 36.072144 counter-clockwise
2003 Empire Maker 1.5 148.3 36.412677 counter-clockwise
2004 Birdstone 1.5 147.5 36.610169 counter-clockwise
2005 Afleet Alex 1.5 148.7 36.314728 counter-clockwise
2006 Jazil 1.5 147.9 36.511156 counter-clockwise
2007 Rags to Riches 1.5 148.7 36.314728 counter-clockwise
2008 Da'Tara 1.5 149.7 36.072144 counter-clockwise
2009 Summer Bird 1.5 147.5 36.610169 counter-clockwise
2010 Drosselmeyer 1.5 151.6 35.620053 counter-clockwise
2011 Ruler on Ice 1.5 150.9 35.785288 counter-clockwise
2012 Union Rags 1.5 150.4 35.904255 counter-clockwise
2013 Palace Malice 1.5 150.7 35.83278 counter-clockwise
2014 Tonalist 1.5 148.5 36.363636 counter-clockwise
2015 American Pharoah 1.5 146.7 36.809816 counter-clockwise
To solve the one-way ANOVA for the given data, follow these steps:
Step 1: Set up the hypothesis
- Null hypothesis: There is no significant difference in the mean speed of winners among different years.
- Alternative hypothesis: There is a significant difference in the mean speed of winners among different years.
Step 2: Calculate the mean for each group (year)
- Calculate the mean speed for each year using the given data.
Step 3: Calculate the total sum of squares (SST)
- Calculate the overall mean speed across all years.
- Calculate the sum of squared differences between each year's mean speed and the overall mean speed.
- SST = sum of (speed - overall mean speed)^2
Step 4: Calculate the between-group sum of squares (SSB) or treatment sum of squares (SSTr)
- Calculate the sum of squared difference between each year's mean speed and the overall mean speed.
- Multiply each squared difference by the number of data points in that year.
- SSB = sum of (number of data points in each year * (year mean speed - overall mean speed)^2)
Step 5: Calculate the within-group sum of squares (SSE) or error sum of squares (SSE)
- Calculate the sum of squared differences between each individual data point and its respective year mean speed.
- SSE = sum of (speed - year mean speed)^2
Step 6: Calculate the degrees of freedom (df)
- df(total) = total number of data points - 1
- df(treatment) = number of groups (years) - 1
- df(error) = df(total) - df(treatment)
Step 7: Calculate the mean square (MS)
- MS(treatment) = SSB / df(treatment)
- MS(error) = SSE / df(error)
Step 8: Calculate the F-value
- F-value = MS(treatment) / MS(error)
Step 9: Determine the critical F-value and compare with the calculated F-value
- Use the significance level (alpha) and the degrees of freedom to find the critical F-value from an F-table.
- If the calculated F-value is greater than the critical F-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 10: Interpret the results
- If the null hypothesis is rejected, conclude that there is a significant difference in the mean speed of winners among different years. If the null hypothesis is not rejected, conclude that there is no significant difference in the mean speed of winners among different years.
To solve the one-way ANOVA for the given data, you can follow these steps:
Step 1: Identify the variables and their levels.
In this case, the variable of interest is "Speed" and the levels are the different years.
Step 2: Calculate the mean speed for each year.
Calculate the mean speed for each year by summing up the speeds and dividing by the number of observations in each year.
Step 3: Calculate the overall mean.
Calculate the overall mean speed by summing up all the speeds and dividing by the total number of observations.
Step 4: Calculate the sum of squares for between groups (SSB).
The sum of squares for between groups measures the variation in mean speeds across different years. It is calculated by the formula:
SSB = Σ(ni * (mean of group i - overall mean)^2)
where ni is the number of observations in group i.
Step 5: Calculate the sum of squares for within groups (SSW).
The sum of squares for within groups measures the variation within each group of speeds. It is calculated by the formula:
SSW = Σ((speed - mean of group)^2)
where speed is the individual speed measurement within each group.
Step 6: Calculate the degrees of freedom (DFB and DFW).
DFB is the degrees of freedom for between groups and is equal to the number of groups minus 1.
DFW is the degrees of freedom for within groups and is equal to the total number of observations minus the number of groups.
Step 7: Calculate the mean squares (MSB and MSW).
MSB is the mean square for between groups and is calculated by dividing SSB by DFB.
MSW is the mean square for within groups and is calculated by dividing SSW by DFW.
Step 8: Calculate the F-value.
The F-value is the ratio of MSB to MSW and is calculated by dividing MSB by MSW.
Step 9: Determine the critical value and compare it to the F-value.
You would need to consult an F-distribution table or use a statistical software to determine the critical value. Compare the critical value to the calculated F-value to determine if there is a significant difference in mean speeds across the different years.
Step 10: Interpret the results.
If the calculated F-value is greater than the critical value, you can conclude that there is a significant difference in mean speeds across the different years. If the calculated F-value is not greater than the critical value, you cannot conclude that there is a significant difference in mean speeds.
Note: It is important to check the assumptions for ANOVA, such as normality and equal variance, before applying this method.