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I'm using Highcharts and would like to display a simple column graph, but instead of using numeric values for the y-axis, I would like to use text values.
For example, instead of [0,5,10,15,20] I would like to use [Very Low,Low,Medium,High,Very High].

I noticed it's somewhat possible to do this with plot bands, but that still shows the numeric y-axis labels and just puts the text beside them. I want to only show the text labels.

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You can change the labels by using a label formatter. Assuming your data is formed appropriately, you can do something like the following:

var yourLabels = ["Very Low", "Low", "Medium", "High", "Very High"];
var yourChart = new Highcharts.Chart({
    //...
    yAxis: {        
        labels: {
            formatter: function() {
                return yourLabels[this.value];
            }
        }
    }
    //...
});

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